A short intro to classical ordination methods - in R ;-)
Ecological datasets have one or both of these datasets (if not more):
Ecological datasets have one or both of these datasets (if not more):
The classical ordination methods always target dimension reduction, and
Note that ordination is a word almost exclusively used by ecologists. Outside most of these methods are known as scaling techniques. The phrase gradient analysis makes a bit more sense in ecology ;-)
Specifically PCA and RDA are quite useful outside of the classical species or species~environment framework of ecology.
PCA can be understood as a MLR on theoretical (latent) instead of observed dependent variables.
The regression coefficients in PCA are factor loadings.
The predicted values of the theoretical dependent are scores.
For instance, in a PCA on this matrix of n objects times p variables PC1 is computed as:
\[ PC_1=f_1X_1+f_2X_2+...+f_pX_p \] There must be as many PCs as original variables. The factor loading matrix translates p \(X\)-variables into p \(PCs\).
PCA can be understood as a rotation of the original coordinate system made up of p \(X\)-variables into a new coordinate system defined by p \(PCs\).
library(vegan)
## Loading required package: permute
## Loading required package: lattice
## This is vegan 2.5-6
library(shape) # nice arrows
mara<-read.table(file="data/MaraRiver.txt",header=TRUE) # water chemistry in 54 streams, 3 types of landuse
wc<-log(mara[-c(5,6),9:26]) # delete two cases with NA data, log-transform concentration data
landuse<-mara$landuse[-c(5,6)]
plot(wc) # check potential for PCA by correlation plot
pca<-prcomp(scale(wc),retx=T,center=F,scale.=F)
pca<-prcomp(wc,retx=T,center=T,scale.=T) # equivalent to line above
summary(pca)
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 2.7152 1.9490 1.22235 1.11919 0.88947 0.84667 0.80815
## Proportion of Variance 0.4096 0.2110 0.08301 0.06959 0.04395 0.03982 0.03628
## Cumulative Proportion 0.4096 0.6206 0.70362 0.77321 0.81716 0.85699 0.89327
## PC8 PC9 PC10 PC11 PC12 PC13 PC14
## Standard deviation 0.69942 0.68698 0.57278 0.44077 0.34335 0.32238 0.29800
## Proportion of Variance 0.02718 0.02622 0.01823 0.01079 0.00655 0.00577 0.00493
## Cumulative Proportion 0.92045 0.94667 0.96489 0.97569 0.98224 0.98801 0.99294
## PC15 PC16 PC17 PC18
## Standard deviation 0.26124 0.20738 0.09562 0.08134
## Proportion of Variance 0.00379 0.00239 0.00051 0.00037
## Cumulative Proportion 0.99674 0.99912 0.99963 1.00000
pca$sdev # stdevs of PCs (squares are eigenvalues)
## [1] 2.71519185 1.94904622 1.22234767 1.11918666 0.88947376 0.84666850
## [7] 0.80815199 0.69941920 0.68698308 0.57277816 0.44076721 0.34334955
## [13] 0.32237891 0.29800143 0.26124205 0.20738029 0.09561853 0.08134109
head(scores<-pca$x) # site scores on all PCs
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## 1 -5.170835 4.016387 -0.5991588 1.2909046 -0.1010020 0.16667942 0.6343563
## 2 -4.109584 3.447343 0.1318830 -2.3348590 -0.7292225 -0.47398838 0.3727337
## 3 -3.448190 4.451837 -0.9213255 0.5202531 -0.6210845 0.55669779 -1.1869106
## 4 -4.026131 2.619122 0.3155890 0.2170746 0.2018600 -0.04762554 -0.7586118
## 7 2.232793 1.480272 0.4206951 0.6272105 1.5808947 -0.34864745 -0.5814405
## 8 5.913178 1.424879 -3.2684397 -2.9349723 -1.0034926 1.03815652 -0.2327136
## PC8 PC9 PC10 PC11 PC12 PC13
## 1 -0.2298109 -0.9806756 -0.3871041 0.3571909 -0.04236249 -0.03051565
## 2 -0.1099423 -1.3833508 -1.1315837 -0.2817962 -0.63408704 0.33104112
## 3 -0.6201881 0.1749492 -0.5086871 0.6001108 -0.52214446 0.20520087
## 4 -1.0545257 0.6203206 -0.6279348 0.1036275 0.48580457 -0.21386990
## 7 -0.9861408 1.3964450 -0.5847160 -1.2657650 -0.48722114 -0.36661223
## 8 -0.2793900 0.5258090 0.7129195 -0.1997592 -0.26750430 -0.60819198
## PC14 PC15 PC16 PC17 PC18
## 1 -0.4348322 -0.06170742 0.03812678 -0.06946439 0.04456263
## 2 0.1136519 0.13224238 0.15079191 -0.27204153 -0.18865812
## 3 0.5252942 -0.62886628 -0.05060063 0.10903471 0.06976005
## 4 0.1518358 0.38178030 -0.12215586 -0.07002405 0.07068122
## 7 -0.3610469 0.48773223 0.09650913 0.06186595 0.03892577
## 8 -0.5882226 -0.29698046 -0.07645784 -0.11833869 -0.12134436
head(scale(wc) %*% pca$rotation) # to manually compute scores from variables and loadings
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## 1 -5.170835 4.016387 -0.5991588 1.2909046 -0.1010020 0.16667942 0.6343563
## 2 -4.109584 3.447343 0.1318830 -2.3348590 -0.7292225 -0.47398838 0.3727337
## 3 -3.448190 4.451837 -0.9213255 0.5202531 -0.6210845 0.55669779 -1.1869106
## 4 -4.026131 2.619122 0.3155890 0.2170746 0.2018600 -0.04762554 -0.7586118
## 7 2.232793 1.480272 0.4206951 0.6272105 1.5808947 -0.34864745 -0.5814405
## 8 5.913178 1.424879 -3.2684397 -2.9349723 -1.0034926 1.03815652 -0.2327136
## PC8 PC9 PC10 PC11 PC12 PC13
## 1 -0.2298109 -0.9806756 -0.3871041 0.3571909 -0.04236249 -0.03051565
## 2 -0.1099423 -1.3833508 -1.1315837 -0.2817962 -0.63408704 0.33104112
## 3 -0.6201881 0.1749492 -0.5086871 0.6001108 -0.52214446 0.20520087
## 4 -1.0545257 0.6203206 -0.6279348 0.1036275 0.48580457 -0.21386990
## 7 -0.9861408 1.3964450 -0.5847160 -1.2657650 -0.48722114 -0.36661223
## 8 -0.2793900 0.5258090 0.7129195 -0.1997592 -0.26750430 -0.60819198
## PC14 PC15 PC16 PC17 PC18
## 1 -0.4348322 -0.06170742 0.03812678 -0.06946439 0.04456263
## 2 0.1136519 0.13224238 0.15079191 -0.27204153 -0.18865812
## 3 0.5252942 -0.62886628 -0.05060063 0.10903471 0.06976005
## 4 0.1518358 0.38178030 -0.12215586 -0.07002405 0.07068122
## 7 -0.3610469 0.48773223 0.09650913 0.06186595 0.03892577
## 8 -0.5882226 -0.29698046 -0.07645784 -0.11833869 -0.12134436
head(loadings<-pca$rotation) # the variable loadings
## PC1 PC2 PC3 PC4 PC5 PC6
## cond -0.25722323 0.1457565 -0.3356613 -0.1583734 -0.04126487 -0.40451583
## pH 0.05614221 0.1125082 0.5222340 -0.3741856 -0.17110129 -0.25878527
## turb -0.13203932 -0.3855016 -0.1073370 -0.3716724 -0.22025887 -0.01157911
## Alk -0.13633672 0.3519383 -0.2457525 0.0313204 0.09063681 -0.44809661
## TSS -0.11636881 -0.3402503 -0.1025929 -0.4995496 -0.25110405 0.04368404
## PO4 -0.08826034 -0.2141495 0.4323275 -0.1977583 0.60913902 -0.12539930
## PC7 PC8 PC9 PC10 PC11 PC12
## cond 0.10974673 -0.050428262 0.17413681 0.013940888 0.64808704 -0.08137442
## pH -0.58257813 0.096086825 0.26386503 -0.205980904 0.05773032 -0.05873929
## turb 0.08448051 -0.011737527 -0.06980739 0.001650434 -0.31037110 -0.49543374
## Alk 0.04261110 -0.306722599 0.20931630 -0.141917638 -0.60093241 -0.08602031
## TSS 0.13099111 -0.073853481 -0.15289911 -0.157532776 -0.05344048 0.39788229
## PO4 0.40877114 -0.002124487 0.17867170 -0.021848936 -0.06745641 0.06817771
## PC13 PC14 PC15 PC16 PC17
## cond 0.07842483 -0.11147245 0.19970689 -0.2485760664 -0.059467443
## pH -0.05666025 -0.03676865 -0.04291062 0.0310385860 -0.004463366
## turb 0.38994704 -0.19755992 -0.16242245 -0.2347958749 0.090676524
## Alk -0.18762446 0.11999960 0.01473726 0.0008990899 -0.068827445
## TSS -0.36793380 0.35906128 0.20457551 -0.0570037711 -0.064652896
## PO4 0.03969603 -0.21939294 0.26938989 0.0462139650 -0.019837594
## PC18
## cond 0.118215868
## pH 0.018664958
## turb 0.029801918
## Alk 0.065155143
## TSS 0.009829404
## PO4 -0.007232622
# how many axes should be kept?)
screeplot(pca,npcs=length(pca$sdev),type = "lines") # plots eigenvalues vs. component #
abline(h=1,col="red")
# -> the first 3-4 PCs seem useful, and just PC1 and PC2 alone are already explaining a lot of overall variance
# eigenvalue of PC5<1, so PC5 contributes less than one original variable (Kaiser-Guttman criterion)
# for a DISTANCE BIPLOT (focus is on sites, "scaling 1")
# each principal component has variance given by eigenvalue, loadings remain unscaled
plot(scores[,1:2],asp=1,pch=21,bg=landuse) # note asp=1
arrows<-loadings*7 # with extension factor
Arrows(x0=0,y0=0,x1=arrows[,1],y1=arrows[,2],col="darkgreen")
text(x=arrows[,1]*1.3,y=arrows[,2]*1.2,labels=names(wc),cex=0.7)
# in this plot:
# 1) distances among sites are approximating true Euclidean distances in multivariate space
# 2) angles between arrows do not reflect correlations among variables
# 3) projecting site on descriptor at right angle gives its appr. descriptor value
# for a CORRELATION BIPLOT (focus is on variables, "scaling 2")
# each principal component is weighted by 1/sqrt(eigenvalue), so it has variance 1
var(scores[,1]/pca$sdev[1]) # just demo
## [1] 1
plot(scores[,1]/pca$sdev[1],scores[,2]/pca$sdev[2],pch=21,bg=landuse,asp=1)
# loadings are weighted by sqrt(eigenvalues)
arrows<-loadings*matrix(pca$sdev,nrow=nrow(loadings),ncol=ncol(loadings),byrow=TRUE)
arrows<-arrows*2 # choose extension factor
Arrows(x0=0,y0=0,x1=arrows[,1],y1=arrows[,2],col="purple")
# as alternative just compute correlation of scores with original data ("structure coefficients")
(structure<-cor(wc,scores))
## PC1 PC2 PC3 PC4 PC5 PC6
## cond -0.6984104 0.28408615 -0.41029477 -0.177249446 -0.03670402 -0.342490809
## pH 0.1524369 0.21928376 0.63835148 -0.418783552 -0.15219011 -0.219105339
## turb -0.3585121 -0.75136050 -0.13120308 -0.415970767 -0.19591449 -0.009803667
## Alk -0.3701803 0.68594395 -0.30039498 0.035053376 0.08061906 -0.379389286
## TSS -0.3159637 -0.66316347 -0.12540419 -0.559089276 -0.22335047 0.036985903
## PO4 -0.2396438 -0.41738723 0.52845449 -0.221328417 0.54181317 -0.106171637
## NH4 -0.4654270 -0.02194250 -0.53020009 -0.186343168 0.45612324 0.162850758
## NO2 -0.5188878 -0.52230884 -0.01574993 0.005211432 0.23769161 -0.002141302
## NO3 -0.5730405 -0.65160683 0.03213869 0.382790600 -0.08425720 0.030755129
## Br -0.9313245 -0.07779131 -0.01608737 0.019147954 -0.14787110 0.071294662
## Cl -0.8345065 -0.10268275 0.30020535 0.279338280 0.10033310 -0.071715149
## Fl -0.8480805 -0.07571208 0.09535793 -0.020000354 0.06993200 -0.351671172
## SO4 -0.7986422 0.21769902 0.23421111 0.278802688 -0.11423549 0.046932670
## Na -0.8811191 0.16618532 0.01657297 0.038583410 -0.22572697 -0.089087312
## K -0.7557029 0.41508974 0.24253708 -0.075909350 -0.00631684 0.223742281
## Ca -0.6701345 0.61799716 -0.01657030 -0.233382943 0.03311222 0.221873328
## Mg -0.7070641 0.54017541 0.09521808 -0.211787735 0.00985287 0.346221391
## TDN -0.6378635 -0.61620157 -0.10341460 0.256075454 -0.06798394 0.064818416
## PC7 PC8 PC9 PC10 PC11
## cond 0.08869204 -0.035270495 0.119629039 0.0079850361 0.285655516
## pH -0.47081167 0.067204970 0.181270813 -0.1179813623 0.025445632
## turb 0.06827309 -0.008209452 -0.047956499 0.0009453327 -0.136801402
## Alk 0.03443624 -0.214527675 0.143796760 -0.0812873231 -0.264871300
## TSS 0.10586072 -0.051654543 -0.105039104 -0.0902313332 -0.023554813
## PO4 0.33034921 -0.001485907 0.122744436 -0.0125145935 -0.029732574
## NH4 -0.32682252 0.325631695 -0.037652130 -0.1272200457 -0.032943687
## NO2 -0.36795338 -0.451185072 -0.141524231 0.1951870779 0.047492681
## NO3 -0.05158254 0.000133211 0.277017753 -0.0433056814 -0.016570634
## Br 0.14345062 -0.019023291 -0.004000929 -0.1725441479 0.066895666
## Cl 0.01996508 0.024214082 -0.120525858 -0.1886080680 0.048860822
## Fl 0.03972002 0.187589873 -0.200265391 0.1477822419 -0.005407743
## SO4 -0.07531681 -0.041088827 -0.301983239 -0.1831854495 -0.043669377
## Na -0.08523333 0.179791964 -0.023950118 0.2200084209 -0.091597546
## K 0.07557981 0.121199537 0.072790327 0.2593618792 -0.007213583
## Ca 0.04548095 -0.165843239 0.100610952 -0.0774684592 0.016715739
## Mg 0.01839427 -0.106342119 0.063503614 0.0234741312 -0.013933336
## TDN -0.09445248 0.037539759 0.334130611 0.0077866617 -0.019725516
## PC12 PC13 PC14 PC15 PC16
## cond -0.02793987 0.025282511 -0.0332189485 0.052171838 -0.0515497755
## pH -0.02016811 -0.018266069 -0.0109571105 -0.011210059 0.0064367908
## turb -0.17010695 0.125710701 -0.0588731392 -0.042431573 -0.0486920355
## Alk -0.02953504 -0.060486168 0.0357600539 0.003849992 0.0001864535
## TSS 0.13661271 -0.118614098 0.1070007750 0.053443725 -0.0118214583
## PO4 0.02340879 0.012797165 -0.0653794093 0.070375967 0.0095838652
## NH4 -0.01376245 -0.020251926 -0.0060367305 0.016144920 0.0028392903
## NO2 -0.02645191 -0.020466386 0.0025810284 0.009774220 0.0245358000
## NO3 0.06814171 -0.023152938 -0.0062065455 -0.018023245 -0.0252696403
## Br -0.10337675 -0.088756432 -0.0618928002 -0.030769421 0.1430478324
## Cl -0.07884890 0.093133026 0.2026581905 0.002730477 -0.0151599829
## Fl 0.10531097 -0.019250758 -0.0169508010 -0.162688990 0.0023135739
## SO4 0.03177807 -0.044383121 -0.1385124348 0.063800533 -0.0813112489
## Na 0.05382217 0.106875862 0.0049270489 0.137864470 0.0702378702
## K -0.13408408 -0.158024615 0.0489002643 0.007774656 -0.0586163075
## Ca 0.08575336 0.066801075 -0.0031824718 -0.045231364 -0.0142660407
## Mg 0.06245134 0.090206616 0.0004852794 -0.046904944 0.0073118804
## TDN 0.03187638 -0.003105378 -0.0152069323 -0.017003161 -0.0228256456
## PC17 PC18
## cond -0.0056861896 0.0096158080
## pH -0.0004267806 0.0015182281
## turb 0.0086703562 0.0024241206
## Alk -0.0065811793 0.0052997906
## TSS -0.0061820150 0.0007995345
## PO4 -0.0018968416 -0.0005883094
## NH4 0.0050891691 0.0049022516
## NO2 0.0027370353 -0.0011256331
## NO3 0.0441001155 0.0418426505
## Br 0.0009045142 -0.0009829097
## Cl -0.0034330802 -0.0031772767
## Fl -0.0013409888 -0.0022312652
## SO4 -0.0082417516 -0.0023487676
## Na 0.0102682671 -0.0035812642
## K 0.0081296716 -0.0014049273
## Ca 0.0495490253 -0.0377676853
## Mg -0.0458882844 0.0395256212
## TDN -0.0464454202 -0.0410550878
structure<-2*structure
Arrows(x0=0,y0=0,x1=structure[,1],y1=structure[,2],col="red")
text(x=arrows[,1]*1.3,y=arrows[,2]*1.2,labels=names(wc),cex=0.7)
# in this plot
# 1) distances among sites are not approximating true Euclidean distances in multivariate space
# 2) angles between arrows reflect correlations among variables (NOT proximity of arrow heads)
# 3) projecting site on descriptor at right angle gives its appr. descriptor value
# PCA using the rda function from the vegan package
pca2<-rda(X=wc,scale=TRUE)
summary(pca2,scaling=1)
##
## Call:
## rda(X = wc, scale = TRUE)
##
## Partitioning of correlations:
## Inertia Proportion
## Total 18 1
## Unconstrained 18 1
##
## Eigenvalues, and their contribution to the correlations
##
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Eigenvalue 7.3723 3.7988 1.49413 1.25258 0.79116 0.71685 0.65311
## Proportion Explained 0.4096 0.2110 0.08301 0.06959 0.04395 0.03982 0.03628
## Cumulative Proportion 0.4096 0.6206 0.70362 0.77321 0.81716 0.85699 0.89327
## PC8 PC9 PC10 PC11 PC12 PC13
## Eigenvalue 0.48919 0.47195 0.32807 0.19428 0.117889 0.103928
## Proportion Explained 0.02718 0.02622 0.01823 0.01079 0.006549 0.005774
## Cumulative Proportion 0.92045 0.94667 0.96489 0.97569 0.982236 0.988010
## PC14 PC15 PC16 PC17 PC18
## Eigenvalue 0.088805 0.068247 0.043007 0.0091429 0.0066164
## Proportion Explained 0.004934 0.003792 0.002389 0.0005079 0.0003676
## Cumulative Proportion 0.992944 0.996735 0.999124 0.9996324 1.0000000
##
## Scaling 1 for species and site scores
## * Sites are scaled proportional to eigenvalues
## * Species are unscaled: weighted dispersion equal on all dimensions
## * General scaling constant of scores: 5.504409
##
##
## Species scores
##
## PC1 PC2 PC3 PC4 PC5 PC6
## cond -1.4159 0.80230 -1.84762 -0.87175 -0.22714 -2.22662
## pH 0.3090 0.61929 2.87459 -2.05967 -0.94181 -1.42446
## turb -0.7268 -2.12196 -0.59083 -2.04584 -1.21239 -0.06374
## Alk -0.7505 1.93721 -1.35272 0.17240 0.49890 -2.46651
## TSS -0.6405 -1.87288 -0.56471 -2.74973 -1.38218 0.24045
## PO4 -0.4858 -1.17877 2.37971 -1.08854 3.35295 -0.69025
## NH4 -0.9435 -0.06197 -2.38757 -0.91648 2.82267 1.05873
## NO2 -1.0519 -1.47508 -0.07092 0.02563 1.47093 -0.01392
## NO3 -1.1617 -1.84024 0.14473 1.88265 -0.52142 0.19995
## Br -1.8880 -0.21969 -0.07244 0.09417 -0.91508 0.46350
## Cl -1.6918 -0.28999 1.35187 1.37385 0.62090 -0.46624
## Fl -1.7193 -0.21382 0.42941 -0.09837 0.43277 -2.28630
## SO4 -1.6191 0.61482 1.05469 1.37121 -0.70693 0.30512
## Na -1.7863 0.46933 0.07463 0.18976 -1.39689 -0.57918
## K -1.5320 1.17228 1.09218 -0.37334 -0.03909 1.45461
## Ca -1.3585 1.74532 -0.07462 -1.14783 0.20491 1.44246
## Mg -1.4334 1.52554 0.42878 -1.04162 0.06097 2.25087
## TDN -1.2931 -1.74025 -0.46569 1.25944 -0.42071 0.42140
##
##
## Site scores (weighted sums of species scores)
##
## PC1 PC2 PC3 PC4 PC5 PC6
## 1 -0.93940 0.72967 -0.10885 0.234522 -0.01835 0.030281
## 2 -0.74660 0.62629 0.02396 -0.424180 -0.13248 -0.086111
## 3 -0.62644 0.80878 -0.16738 0.094516 -0.11283 0.101137
## 4 -0.73144 0.47582 0.05733 0.039436 0.03667 -0.008652
## 7 0.40564 0.26892 0.07643 0.113947 0.28721 -0.063340
## 8 1.07426 0.25886 -0.59379 -0.533204 -0.18231 0.188605
## 9 -0.32892 0.25033 -0.45377 0.617791 -0.24044 -0.122356
## 10 -0.57184 0.21769 0.03691 -0.379616 0.08341 -0.294496
## 11 0.13734 0.33630 -0.26902 0.054280 0.12820 -0.050003
## 12 0.78569 0.17837 0.01844 0.115072 -0.02183 0.091719
## 13 -0.22093 -0.36549 -0.16998 -0.099846 0.09512 -0.143509
## 14 1.30998 -0.67125 -0.57884 -0.023233 0.14289 -0.479781
## 15 -0.43427 -0.59259 -0.01832 -0.068894 -0.07069 0.077409
## 16 -0.74333 -0.01680 -0.19101 0.152920 0.30242 -0.080449
## 17 -0.84366 -0.21000 -0.04445 -0.008928 0.07997 0.014100
## 18 -0.48203 -0.01707 -0.11851 -0.044274 -0.16607 -0.277649
## 19 -0.28152 -0.26355 0.12280 -0.173659 -0.05794 0.076838
## 20 0.01911 -0.20131 -0.17851 0.014238 -0.28113 0.127905
## 21 -0.40327 -0.31194 0.14452 -0.048581 -0.05699 0.013663
## 22 -0.18831 -0.29426 0.20016 -0.001522 -0.30260 0.030065
## 23 -0.42134 -0.16552 0.05614 -0.096723 0.05274 0.079374
## 24 -0.67474 -0.45065 -0.01582 -0.035151 0.15207 0.097047
## 25 -0.07187 -0.38842 0.37212 -0.278347 -0.02696 -0.088559
## 26 0.16221 -0.46968 0.08605 -0.082842 0.04253 0.100862
## 27 0.02137 -0.26587 -0.13515 0.126573 -0.12436 0.198872
## 28 0.04144 -0.47296 -0.01313 -0.042046 0.11692 0.111934
## 29 0.04714 -0.40758 0.03664 0.034209 0.05919 0.114249
## 30 -0.18686 -0.40929 -0.27928 0.050715 -0.07947 0.204339
## 31 0.37961 0.30976 0.11739 0.216706 -0.20004 -0.433454
## 32 0.17514 0.28779 -0.02441 0.098778 0.05891 0.112364
## 33 0.05528 -0.03829 0.12601 0.556220 0.21520 -0.184383
## 34 -0.12436 0.23716 0.22438 -0.342063 0.07600 -0.046932
## 35 -0.05549 -0.18388 0.16587 -0.045118 0.03482 -0.030455
## 36 0.20401 0.36785 -0.01204 -0.020236 -0.01134 0.003642
## 37 0.54387 0.14867 -0.04013 0.095238 -0.18598 0.179905
## 38 0.35917 0.14769 0.17737 -0.004264 0.22119 0.093820
## 39 0.42440 0.35483 0.14198 0.045518 0.09675 0.190421
## 40 0.33889 0.19385 0.14972 -0.003335 0.25193 0.134582
## 41 0.71629 0.08022 0.68924 0.002355 -0.12205 -0.199364
## 42 0.78687 0.25324 0.02979 0.145646 -0.20004 0.072693
## 43 0.22158 0.29242 0.32534 -0.093631 0.06309 -0.015564
## 44 0.61384 0.01066 0.13364 0.126831 0.28036 0.112728
## 45 0.43087 0.68355 -0.25570 -0.420474 0.16347 -0.019426
## 46 -0.34968 -0.38830 -0.17466 -0.086047 0.04080 -0.195030
## 47 -0.12295 -0.09128 -0.05881 -0.061097 0.09589 -0.145553
## 48 0.21811 0.18380 0.03419 0.164258 0.12432 0.150992
## 49 -0.11584 -0.02878 -0.15463 -0.137653 -0.15426 0.067831
## 50 -0.16384 -0.28048 -0.04759 0.027258 -0.12822 0.077030
## 51 -0.28502 -0.08784 0.14403 0.097088 -0.02838 -0.002284
## 52 0.43786 -0.05135 0.34622 0.043325 -0.44642 -0.131500
## 53 0.11174 -0.24041 -0.02979 0.099997 0.10370 0.125014
## 54 0.09225 -0.33766 0.09691 0.187527 -0.05459 0.119429
scores(wc,scaling=0)
## cond pH turb Alk TSS PO4 NH4
## 1 5.434159 2.008214 -3.9120230 7.677864 2.529802 -14.47716 -10.688750
## 2 5.324959 2.083185 3.5695327 7.677864 4.085042 -13.99943 -10.654264
## 3 5.326419 2.064328 -3.9120230 7.677864 2.358155 -15.80328 -10.490475
## 4 5.202907 2.074429 -3.9120230 7.677864 3.294588 -13.81551 -10.785377
## 7 4.169761 2.076938 -3.9120230 7.677864 2.143157 -13.03161 -11.155251
## 8 4.513055 2.047693 2.0725428 6.703188 3.933333 -15.80328 -10.892349
## 9 5.178407 1.983756 -3.9120230 7.677864 2.208274 -15.80328 -11.381897
## 10 5.139322 2.063058 3.6095656 7.377759 4.163337 -12.89523 -10.785377
## 11 4.967728 2.008214 -3.9120230 7.200425 2.990034 -13.88058 -10.892349
## 12 3.916015 2.060514 -3.9120230 6.131226 2.472806 -14.93331 -11.512925
## 13 4.808111 2.010895 4.7621739 6.866933 4.672829 -12.93174 -10.892349
## 14 4.695925 2.018895 3.4011974 6.429719 3.934002 -14.14959 -11.258283
## 15 5.060694 2.041220 5.3303004 5.247024 4.704110 -13.40340 -10.346655
## 16 5.270946 2.013569 -0.1392621 7.146772 3.107850 -13.24453 -9.903488
## 17 5.141079 2.032088 4.5752263 6.927558 4.064862 -12.90725 -10.542147
## 18 5.015291 2.039921 3.6975914 7.098376 4.098021 -13.70218 -10.819778
## 19 4.581902 2.073172 4.9126549 6.476972 4.478278 -13.22772 -11.205441
## 20 4.374498 2.034706 4.1415462 6.476972 4.322603 -14.87106 -11.061850
## 21 4.774913 2.080691 3.1884166 6.253829 4.028095 -13.11738 -11.061850
## 22 4.632785 2.043814 4.5212450 6.291569 4.479401 -13.07357 -12.764689
## 23 4.745888 2.060514 3.1822118 6.672033 4.055124 -13.10256 -10.752120
## 24 4.827513 2.032088 6.9077553 6.040255 3.898134 -12.93588 -9.875872
## 25 4.305416 2.155245 4.1407509 6.109248 4.365588 -12.99573 -10.855405
## 26 4.097672 2.042518 3.7232809 5.991465 4.392179 -13.05470 -11.322305
## 27 4.332048 2.020222 3.3214324 6.214608 3.647207 -14.45607 -11.107460
## 28 4.198705 2.028148 4.7492705 6.086775 4.188276 -13.05470 -10.654264
## 29 4.239887 2.014903 4.1141472 6.131226 3.943330 -13.00015 -11.445267
## 30 4.461300 1.987874 3.3690185 6.345636 4.820532 -14.32634 -10.785377
## 31 4.919251 2.064328 -3.9120230 7.138867 2.580217 -14.12068 -12.764689
## 32 4.427239 2.041220 -3.9120230 6.606650 2.782952 -14.21004 -10.976432
## 33 4.112512 2.042518 -3.9120230 6.866933 1.798029 -13.86891 -10.976432
## 34 4.750136 2.096790 2.0175661 6.917706 3.767660 -12.93174 -11.258283
## 35 4.443827 2.081938 1.1052568 6.429719 3.673766 -12.84473 -10.976432
## 36 4.517431 2.064328 -3.9120230 6.672033 2.953935 -14.39891 -10.892349
## 37 4.091006 2.043814 -3.9120230 6.253829 3.506558 -15.11014 -11.754997
## 38 4.169761 2.063058 -3.9120230 6.345636 2.907440 -13.05939 -11.018229
## 39 4.320151 2.036012 -3.9120230 6.429719 2.471208 -13.23890 -11.849798
## 40 4.215086 2.051556 -3.9120230 6.363028 2.849778 -12.94422 -10.892349
## 41 3.949319 2.143589 -3.9120230 6.152733 2.625358 -13.16319 -14.152383
## 42 3.931826 2.104134 -3.9120230 6.194405 2.330756 -15.80328 -11.205441
## 43 4.418841 2.078191 -3.9120230 6.620073 3.224267 -12.94841 -12.206073
## 44 3.919991 2.028148 -3.9120230 6.152733 2.768515 -13.08314 -11.445267
## 45 5.196838 2.075684 -3.9120230 7.265430 2.644681 -14.03865 -10.625034
## 46 4.910447 2.038620 4.7405748 6.856462 4.499810 -13.47192 -10.654264
## 47 4.680278 2.048982 2.5610958 6.756932 3.642359 -13.66709 -10.688750
## 48 4.300003 2.013569 -3.9120230 6.646391 2.870044 -13.81551 -11.849798
## 49 4.565389 2.039921 4.6293749 6.756932 4.124881 -14.45607 -10.936312
## 50 4.527209 2.041220 3.6041382 6.363028 3.964469 -13.78595 -11.155251
## 51 4.647271 2.064328 1.4906544 6.507278 3.210844 -13.14768 -11.107460
## 52 4.237001 2.091864 1.8381658 6.309918 3.332205 -14.36196 -14.152383
## 53 4.289089 2.012233 4.2668963 6.214608 3.036554 -13.06882 -10.855405
## 54 4.262680 1.991976 3.4177267 6.152733 3.463541 -13.11738 -12.764689
## NO2 NO3 Br Cl Fl SO4 Na
## 1 -13.93092 -9.817310 -12.91942 -8.235041 -10.015537 -9.363492 -6.979470
## 2 -14.31309 -11.760106 -13.09280 -8.344998 -10.149388 -9.885648 -7.078470
## 3 -13.69329 -9.959000 -13.48621 -8.417488 -10.458613 -10.243165 -7.382206
## 4 -13.04078 -8.989071 -13.60040 -8.440737 -10.404363 -9.973910 -7.241213
## 7 -13.83776 -9.856604 -14.90022 -9.743071 -11.205441 -11.198115 -8.360685
## 8 -15.08136 -12.534577 -15.48682 -12.116232 -11.493123 -12.562748 -8.835946
## 9 -14.17935 -7.819967 -13.65000 -9.052858 -10.504968 -10.710924 -7.216567
## 10 -13.22772 -10.747458 -13.93317 -9.000971 -9.664471 -10.756803 -6.930678
## 11 -14.93637 -10.268771 -14.41153 -9.585761 -10.658510 -11.212821 -8.068873
## 12 -14.50866 -10.988197 -15.48682 -10.015537 -10.953310 -11.381897 -8.319154
## 13 -13.00903 -8.886570 -14.11932 -9.243897 -10.191170 -10.876349 -7.939656
## 14 -13.69329 -10.315977 -15.79601 -10.268771 -11.273909 -12.639937 -9.511445
## 15 -12.90725 -8.056369 -13.66709 -8.791788 -10.183201 -10.710924 -7.496422
## 16 -11.45466 -8.514038 -13.58440 -8.667017 -10.035877 -10.277454 -7.376421
## 17 -11.94988 -7.789553 -13.55315 -8.507312 -10.223693 -10.306955 -7.103282
## 18 -14.75456 -8.972789 -13.77629 -8.917395 -9.833961 -10.608707 -6.761265
## 19 -13.12236 -8.561866 -13.93317 -8.917186 -10.538366 -10.785377 -7.908529
## 20 -14.38644 -8.900667 -14.03740 -9.217365 -10.804890 -10.919599 -7.757987
## 21 -13.06882 -7.875745 -13.72933 -8.757489 -10.424364 -10.970601 -7.420468
## 22 -14.81247 -8.104240 -13.90544 -8.823467 -10.576832 -10.897740 -7.684699
## 23 -12.93588 -8.428715 -13.80556 -8.748917 -10.508624 -10.824791 -7.667536
## 24 -11.73357 -8.278432 -13.42347 -8.390292 -10.172675 -10.650036 -7.204263
## 25 -13.00015 -8.675515 -14.17076 -8.934377 -10.427736 -11.049191 -7.918280
## 26 -13.21119 -9.123337 -14.20707 -9.147300 -10.697561 -11.408565 -8.190274
## 27 -13.13241 -8.746267 -14.18948 -9.421061 -10.850237 -10.860600 -7.776376
## 28 -13.21119 -8.933383 -14.24476 -9.052858 -10.620927 -11.258283 -8.086106
## 29 -13.25020 -8.917738 -14.22649 -9.115239 -10.645825 -11.155251 -8.057136
## 30 -12.46744 -8.574171 -13.91976 -9.107829 -10.693146 -10.988197 -7.688587
## 31 -15.85573 -10.417652 -14.45796 -9.303553 -10.549751 -11.000102 -7.930663
## 32 -14.27755 -10.319003 -14.64105 -9.451139 -10.738198 -10.804890 -7.850380
## 33 -13.42347 -8.594868 -15.15111 -8.376106 -10.231992 -10.850237 -7.661326
## 34 -13.77629 -10.596635 -14.53295 -9.064977 -10.504968 -10.724468 -7.819272
## 35 -14.46752 -8.523810 -14.13572 -9.182054 -10.444772 -11.169336 -7.821006
## 36 -14.81247 -10.693146 -14.90022 -9.672376 -10.545942 -11.012150 -7.574122
## 37 -14.75456 -10.309953 -14.97706 -9.772459 -11.055501 -11.068240 -8.167939
## 38 -14.55366 -10.592643 -14.55785 -9.361163 -10.780558 -11.134489 -8.384587
## 39 -15.85573 -10.534599 -14.58554 -9.593066 -10.947612 -11.205441 -8.477747
## 40 -14.93637 -10.604667 -14.69982 -9.342730 -10.780558 -11.127663 -8.383826
## 41 -13.47192 -10.633299 -15.15111 -9.664471 -10.809828 -11.250561 -8.369463
## 42 -14.75456 -10.795086 -15.48682 -10.099502 -11.155251 -10.860600 -8.109847
## 43 -14.65022 -10.637457 -14.48299 -9.198372 -10.629158 -11.055501 -8.135051
## 44 -13.65851 -10.804890 -15.48682 -9.588677 -10.976432 -11.107460 -8.370965
## 45 -15.16258 -11.963911 -15.55848 -10.497695 -10.988197 -12.649240 -7.763909
## 46 -12.41433 -8.320989 -14.03740 -9.217365 -9.997798 -11.000102 -7.744840
## 47 -13.16841 -9.582854 -14.45796 -9.179045 -10.104380 -10.845096 -7.864545
## 48 -13.56865 -10.075463 -14.61180 -9.317926 -10.742817 -10.994132 -8.351429
## 49 -13.67575 -9.648845 -14.18948 -9.348454 -10.569020 -10.834892 -7.567454
## 50 -13.61666 -8.289480 -13.87845 -9.400291 -10.534599 -10.886987 -7.656624
## 51 -14.27755 -8.300913 -13.86575 -8.940603 -10.391248 -10.625034 -7.519088
## 52 -13.74785 -9.777736 -14.90022 -9.729534 -10.804890 -11.024345 -7.768115
## 53 -14.69982 -8.989977 -14.32467 -9.171418 -10.671358 -11.227747 -8.046346
## 54 -13.30269 -8.843859 -14.18948 -9.161169 -10.701995 -11.155251 -8.055215
## K Ca Mg TDN
## 1 -7.766051 -8.022238 -8.787134 -9.456241
## 2 -8.035378 -8.037332 -8.829236 -10.349775
## 3 -7.891573 -7.896082 -8.755051 -9.482149
## 4 -8.113042 -8.110599 -8.928128 -8.818298
## 7 -9.054748 -9.217365 -10.089817 -9.600424
## 8 -9.348454 -9.288302 -10.136681 -10.701995
## 9 -8.771460 -9.145096 -9.938079 -7.789645
## 10 -7.996265 -8.749182 -9.514152 -10.031321
## 11 -9.045818 -8.922421 -9.828380 -9.833961
## 12 -9.387078 -9.686765 -10.280365 -10.504968
## 13 -9.086107 -9.401501 -10.061312 -8.746606
## 14 -11.963911 -10.662775 -11.917891 -9.963238
## 15 -8.472746 -9.585761 -10.104380 -7.951879
## 16 -8.471646 -8.706074 -9.571310 -8.250990
## 17 -8.301493 -8.613313 -9.351904 -7.712952
## 18 -8.898417 -9.145564 -9.841452 -8.825078
## 19 -8.588147 -8.941879 -9.685156 -8.482792
## 20 -9.073313 -9.331379 -10.054310 -8.785073
## 21 -8.607972 -9.059570 -9.746484 -7.829059
## 22 -8.717060 -9.076448 -9.811820 -8.095199
## 23 -8.449292 -8.802498 -9.545813 -8.326573
## 24 -8.326965 -9.201351 -9.706277 -8.069307
## 25 -8.946546 -9.473005 -10.035877 -8.558015
## 26 -9.114512 -9.636519 -10.207299 -9.003326
## 27 -8.958205 -9.444798 -9.986869 -8.645027
## 28 -8.954838 -9.630412 -10.207299 -8.759265
## 29 -8.921904 -9.547213 -10.089817 -8.831904
## 30 -8.852666 -9.322390 -10.004413 -8.454218
## 31 -8.497768 -9.963238 -10.881654 -10.322038
## 32 -8.896355 -8.982966 -9.774215 -9.889585
## 33 -8.780566 -9.866192 -10.448215 -8.497391
## 34 -8.427544 -8.636236 -9.452412 -10.154516
## 35 -8.832308 -9.254292 -9.984698 -8.440232
## 36 -8.667847 -9.083672 -9.854697 -10.085009
## 37 -9.226470 -9.334770 -10.104380 -10.089817
## 38 -8.934050 -9.242864 -9.997798 -10.077841
## 39 -8.815283 -8.916455 -9.636519 -10.292096
## 40 -8.841816 -9.161836 -9.850895 -10.038162
## 41 -9.262687 -9.658191 -10.292096 -10.549751
## 42 -9.575624 -9.761988 -10.340443 -10.274551
## 43 -8.687970 -8.943601 -9.701363 -10.434516
## 44 -9.258481 -9.656627 -10.237563 -10.343544
## 45 -8.333543 -8.629517 -9.376395 -10.384754
## 46 -8.981740 -9.350753 -10.038162 -8.213392
## 47 -8.860388 -9.185160 -10.097072 -9.276480
## 48 -8.773105 -9.000209 -9.726179 -9.893537
## 49 -8.880598 -8.988542 -9.711216 -9.390664
## 50 -8.829858 -9.192726 -9.991226 -8.228262
## 51 -8.589931 -9.069275 -9.819146 -8.239561
## 52 -9.198847 -9.578510 -10.170061 -9.746484
## 53 -8.908009 -9.510095 -10.063656 -8.845697
## 54 -8.846253 -9.446063 -10.038162 -8.811564
biplot(pca2,scaling=1)
# note different scaling factors, but solution remains same
##############################
# some follow-up suggestions #
# test PCA-axes for effect of landuse or stream size (as log(Q)) using ANOVA or ANCOVA
# correlate PCA-axes with other potential "controlling" variables (e.g. TDN, canopy cover) to give "meta-dimensions" more meaning
# useful function envfit() to relate additional variables to the ordination spaceTwo involved matrices: one dependent, one independent.
(Note: For correlation of two matrices see CCorA (Canonical correlation analysis) \(\neq\) CCA.)
Redundancy: The proportion of total variance of in the response variables that can be explained linear combinations of predictors.
Two steps:
The response variables are constrained to be linear combinations of the predictors first! The PCA can only “ordinate” variation of the responses that is relatable to predictors.
Significance tests for the overall model, for the various RDA-axes and for the individual predictors are available (permutation-based).
Site scores:
zwc<-scale(wc) # must at least be centered even if dimensionally homogeneous!
xmat<-data.frame(logQ=log(mara$Q),logTDN=log(mara$TDN),canopy=mara$canopy)[-c(5,6),]
rda<-rda(zwc~logQ+logTDN+canopy,data=xmat) # take care: confusing X and Y argument names
# actual RDA output check
summary(rda)
##
## Call:
## rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
##
## Partitioning of variance:
## Inertia Proportion
## Total 18.000 1.0000
## Constrained 6.217 0.3454
## Unconstrained 11.783 0.6546
##
## Eigenvalues, and their contribution to the variance
##
## Importance of components:
## RDA1 RDA2 RDA3 PC1 PC2 PC3 PC4
## Eigenvalue 5.0745 0.94540 0.19681 4.9144 1.75906 1.00857 0.84980
## Proportion Explained 0.2819 0.05252 0.01093 0.2730 0.09773 0.05603 0.04721
## Cumulative Proportion 0.2819 0.33444 0.34537 0.6184 0.71612 0.77216 0.81937
## PC5 PC6 PC7 PC8 PC9 PC10 PC11
## Eigenvalue 0.70352 0.69192 0.52705 0.43646 0.29351 0.178515 0.115065
## Proportion Explained 0.03908 0.03844 0.02928 0.02425 0.01631 0.009917 0.006392
## Cumulative Proportion 0.85845 0.89689 0.92617 0.95042 0.96672 0.976642 0.983035
## PC12 PC13 PC14 PC15 PC16 PC17
## Eigenvalue 0.096526 0.080421 0.066249 0.041010 0.0137298 0.0074400
## Proportion Explained 0.005363 0.004468 0.003681 0.002278 0.0007628 0.0004133
## Cumulative Proportion 0.988397 0.992865 0.996546 0.998824 0.9995867 1.0000000
##
## Accumulated constrained eigenvalues
## Importance of components:
## RDA1 RDA2 RDA3
## Eigenvalue 5.0745 0.9454 0.19681
## Proportion Explained 0.8163 0.1521 0.03166
## Cumulative Proportion 0.8163 0.9683 1.00000
##
## Scaling 2 for species and site scores
## * Species are scaled proportional to eigenvalues
## * Sites are unscaled: weighted dispersion equal on all dimensions
## * General scaling constant of scores: 5.504409
##
##
## Species scores
##
## RDA1 RDA2 RDA3 PC1 PC2 PC3
## cond -0.57654 0.612080 -0.021819 -6.413e-01 -2.623e-01 -2.670e-01
## pH 0.45481 -0.063413 -0.134308 -1.950e-01 5.990e-01 -4.274e-02
## turb -0.83125 0.001573 0.188277 3.949e-01 5.850e-01 -4.849e-01
## Alk 0.03483 0.465971 -0.022713 -7.910e-01 -5.520e-01 -3.544e-02
## TSS -0.70702 0.219654 0.087775 4.023e-01 6.509e-01 -5.401e-01
## PO4 -0.31467 -0.045169 -0.308413 6.067e-02 9.303e-01 2.458e-01
## NH4 -0.53863 0.291317 -0.065288 -2.359e-01 -2.399e-01 -6.823e-01
## NO2 -0.80583 -0.193545 0.279175 -4.976e-02 3.768e-01 -1.452e-01
## NO3 -1.15497 -0.469987 -0.153789 4.834e-02 7.215e-02 2.538e-01
## Br -0.92421 0.148249 -0.054180 -7.170e-01 1.920e-01 -2.714e-02
## Cl -0.79416 0.002213 0.010841 -7.045e-01 3.496e-01 4.859e-01
## Fl -0.72110 0.145664 -0.007607 -7.091e-01 4.037e-01 4.306e-02
## SO4 -0.49164 0.014945 0.139145 -9.938e-01 1.553e-01 3.314e-01
## Na -0.70235 0.088094 0.050389 -9.311e-01 5.674e-02 -8.230e-02
## K -0.32852 0.130777 -0.117953 -1.089e+00 1.871e-01 -7.620e-02
## Ca -0.19361 0.501576 -0.081789 -1.047e+00 -6.644e-02 -2.463e-01
## Mg -0.23802 0.339320 -0.048151 -1.079e+00 6.218e-02 -2.512e-01
## TDN -1.22312 -0.408704 -0.142092 -4.770e-18 7.749e-17 -5.248e-17
##
##
## Site scores (weighted sums of species scores)
##
## RDA1 RDA2 RDA3 PC1 PC2 PC3
## 1 -0.89854 3.30672 -0.386506 -1.989464 -0.77705 0.7371045
## 2 -0.56847 4.04439 0.534394 -1.743560 0.92041 -0.8501656
## 3 -0.35983 3.03931 -0.034654 -1.545136 -1.28647 0.0427813
## 4 -0.77641 2.20741 -1.483632 -1.559658 -0.36335 -0.0291313
## 7 0.88348 -0.35314 -1.973257 -0.122169 -0.59904 0.1641181
## 8 1.89546 0.74944 2.486365 0.944714 -1.12313 -3.3055183
## 9 -0.52003 0.37308 0.198164 -0.477278 -2.80260 0.6379405
## 10 -0.63914 2.46873 0.151441 -0.963658 1.36394 -0.8199541
## 11 0.46308 1.27858 -1.064719 -0.391269 -0.86740 -0.4922014
## 12 1.44680 -1.16870 1.489863 0.041605 -0.33977 -0.3592235
## 13 -0.69258 0.08168 0.925485 0.819920 0.30530 0.3286962
## 14 1.41198 -2.32777 3.957086 3.261487 -0.87884 0.5640433
## 15 -1.23166 -0.83951 0.486820 0.463530 0.68386 -0.3343295
## 16 -1.26703 1.08821 0.262397 -0.531979 -0.42709 0.1677873
## 17 -1.58220 0.73379 -0.224828 -0.774707 0.18625 -0.6653627
## 18 -0.81300 1.11939 -0.295492 -0.233557 0.01539 0.1495812
## 19 -0.66646 -0.18332 -0.290191 0.319756 0.60538 0.1107652
## 20 -0.20474 -0.47474 1.516958 0.555638 -0.47108 -0.1279505
## 21 -0.93565 -0.55098 -1.434015 -0.025914 0.41162 -0.2975757
## 22 -0.56437 -0.70216 -1.466515 0.227191 0.38038 0.5203794
## 23 -0.83147 0.22256 -0.971790 -0.039558 0.27401 -0.0556500
## 24 -1.48587 -0.22424 0.743449 -0.289558 0.79299 -0.7436195
## 25 -0.37834 -1.18949 -1.029574 0.298653 1.42737 -0.4827891
## 26 -0.12196 -1.44806 0.564615 0.712294 0.85053 -0.1981804
## 27 -0.25191 -1.07885 1.937866 0.640716 -0.48858 0.2038573
## 28 -0.34372 -1.24137 0.490924 0.638891 0.55613 -0.3185752
## 29 -0.27388 -1.25221 0.563976 0.447446 0.47782 0.0534905
## 30 -0.72988 -0.59341 2.637512 0.500231 -0.27123 -0.4969936
## 31 0.90276 -0.02438 -0.951023 -0.098103 -0.24096 1.4932089
## 32 0.53785 0.31010 -0.000380 -0.195213 -0.44407 0.2430924
## 33 0.02874 -1.71644 -0.584641 -0.191690 -0.82265 1.3993386
## 34 0.10795 1.36339 -0.679776 -0.140368 1.14453 0.2044740
## 35 -0.24197 -0.66292 -2.716238 0.094133 0.30400 -0.1579230
## 36 0.67101 0.64767 -0.370543 -0.390286 -0.26097 -0.3706464
## 37 1.00078 -0.66312 1.406747 0.710440 -0.51072 0.5775779
## 38 0.77136 -0.28513 -1.624952 0.022634 0.37933 0.2749370
## 39 1.03318 0.20503 -2.567599 -0.054682 -0.06822 0.5670135
## 40 0.77241 -0.06114 -2.117887 -0.065082 0.24383 0.1888934
## 41 1.38747 -1.92709 -0.032796 0.002718 1.44558 1.2661054
## 42 1.49551 -1.30131 2.018641 0.082048 -0.65703 -0.4330951
## 43 0.71385 0.41829 -1.772975 -0.186986 0.86709 0.8222904
## 44 1.06214 -1.26145 0.642918 0.239372 0.27914 0.4771499
## 45 1.26992 2.30778 -2.592650 0.197937 -1.09176 -1.0417050
## 46 -0.94288 -0.05952 0.930304 -0.120353 0.25364 -1.2621277
## 47 -0.26421 0.23181 0.880294 -0.200536 0.30144 -0.4244519
## 48 0.52678 -0.11986 0.294289 0.069655 -0.25116 0.8955469
## 49 -0.22617 0.54673 1.879041 -0.101247 0.12653 -0.9326487
## 50 -0.55083 -0.68713 -0.007739 0.367544 -0.20260 -0.0379831
## 51 -0.54615 -0.33780 -2.156689 -0.035535 -0.10177 0.6969878
## 52 0.72450 -1.54996 2.095876 0.096846 0.56095 0.7707365
## 53 -0.05122 -0.96640 -1.060389 0.344503 -0.07383 0.0001981
## 54 -0.14642 -1.49251 0.796025 0.367647 0.26392 0.6797064
##
##
## Site constraints (linear combinations of constraining variables)
##
## RDA1 RDA2 RDA3 PC1 PC2 PC3
## 1 -0.196294 0.827168 1.024793 -1.989464 -0.77705 0.7371045
## 2 0.456938 1.205241 1.334894 -1.743560 0.92041 -0.8501656
## 3 -0.155055 0.937232 0.556790 -1.545136 -1.28647 0.0427813
## 4 -0.350139 -0.151953 0.153012 -1.559658 -0.36335 -0.0291313
## 7 0.645108 -0.731268 -0.602499 -0.122169 -0.59904 0.1641181
## 8 1.399964 -0.094265 -0.277382 0.944714 -1.12313 -3.3055183
## 9 -0.949046 -0.756474 -1.035017 -0.477278 -2.80260 0.6379405
## 10 0.400897 1.170318 -0.584368 -0.963658 1.36394 -0.8199541
## 11 0.683071 -0.182817 -0.671894 -0.391269 -0.86740 -0.4922014
## 12 1.415316 -1.241293 1.341637 0.041605 -0.33977 -0.3592235
## 13 -0.900162 1.328320 0.066534 0.819920 0.30530 0.3286962
## 14 0.080004 1.535836 0.591574 3.261487 -0.87884 0.5640433
## 15 -1.179265 -0.030352 0.132810 0.463530 0.68386 -0.3343295
## 16 -1.033267 0.516938 -0.347991 -0.531979 -0.42709 0.1677873
## 17 -1.121367 -0.744455 -0.188840 -0.774707 0.18625 -0.6653627
## 18 -0.662565 0.751154 0.297991 -0.233557 0.01539 0.1495812
## 19 -0.920492 0.457760 0.672744 0.319756 0.60538 0.1107652
## 20 -0.586320 0.202346 0.905904 0.555638 -0.47108 -0.1279505
## 21 -0.911546 -0.594593 -1.513758 -0.025914 0.41162 -0.2975757
## 22 -0.758713 -0.515269 -0.966421 0.227191 0.38038 0.5203794
## 23 -0.928113 0.205313 0.237043 -0.039558 0.27401 -0.0556500
## 24 -0.903223 -0.450166 -0.113183 -0.289558 0.79299 -0.7436195
## 25 -0.368012 -0.496738 -0.746479 0.298653 1.42737 -0.4827891
## 26 -0.089848 -0.322454 -0.143368 0.712294 0.85053 -0.1981804
## 27 -0.697730 0.173662 0.847072 0.640716 -0.48858 0.2038573
## 28 -0.346198 -0.336436 0.185893 0.638891 0.55613 -0.3185752
## 29 -0.263150 -0.581729 0.747294 0.447446 0.47782 0.0534905
## 30 -0.791706 -0.061225 0.832429 0.500231 -0.27123 -0.4969936
## 31 0.923261 0.244046 -0.132397 -0.098103 -0.24096 1.4932089
## 32 0.493747 0.270364 0.091327 -0.195213 -0.44407 0.2430924
## 33 -0.228132 -1.320830 -0.056533 -0.191690 -0.82265 1.3993386
## 34 0.325305 1.783462 -0.729323 -0.140368 1.14453 0.2044740
## 35 -0.351369 -0.719932 -1.173188 0.094133 0.30400 -0.1579230
## 36 0.816178 0.094178 -0.641906 -0.390286 -0.26097 -0.3706464
## 37 0.562113 0.968540 -0.932109 0.710440 -0.51072 0.5775779
## 38 0.838366 -0.001746 -0.613313 0.022634 0.37933 0.2749370
## 39 1.028180 0.325781 -1.505883 -0.054682 -0.06822 0.5670135
## 40 0.826971 -0.078181 -0.607127 -0.065082 0.24383 0.1888934
## 41 1.462892 -0.923351 0.369467 0.002718 1.44558 1.2661054
## 42 1.320082 -1.062884 -0.162151 0.082048 -0.65703 -0.4330951
## 43 0.949646 0.830000 -1.161163 -0.186986 0.86709 0.8222904
## 44 1.047701 -0.638899 1.505043 0.239372 0.27914 0.4771499
## 45 0.483612 1.565549 0.343759 0.197937 -1.09176 -1.0417050
## 46 -0.529654 -1.122990 -0.261496 -0.120353 0.25364 -1.2621277
## 47 -0.008242 -0.330815 1.324407 -0.200536 0.30144 -0.4244519
## 48 0.420789 0.528903 0.006754 0.069655 -0.25116 0.8955469
## 49 0.094762 0.175222 -0.120623 -0.101247 0.12653 -0.9326487
## 50 -0.887653 -0.216708 0.330209 0.367544 -0.20260 -0.0379831
## 51 -0.864402 -0.088076 -0.151145 -0.035535 -0.10177 0.6969878
## 52 0.586747 -0.983296 1.772374 0.096846 0.56095 0.7707365
## 53 -0.136695 -0.620761 -0.120578 0.344503 -0.07383 0.0001981
## 54 -0.143291 -0.697375 -0.111622 0.367647 0.26392 0.6797064
##
##
## Biplot scores for constraining variables
##
## RDA1 RDA2 RDA3 PC1 PC2 PC3
## logQ 0.5159 -0.6932 0.5034 0 0 0
## logTDN -0.9427 -0.3150 -0.1095 0 0 0
## canopy 0.4907 -0.1427 -0.8596 0 0 0
?cca.object
RsquareAdj(rda) # redundancy statistic (fractional amount of variation of the response data matrix explained by constraints)
## $r.squared
## [1] 0.3453742
##
## $adj.r.squared
## [1] 0.3044601
# hypothesis tests #
# testing the first axis (global test)
anova(rda)
## Permutation test for rda under reduced model
## Permutation: free
## Number of permutations: 999
##
## Model: rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
## Df Variance F Pr(>F)
## Model 3 6.2167 8.4414 0.001 ***
## Residual 48 11.7833
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(rda,first=TRUE)
## Permutation test for rda under reduced model
## Permutation: free
## Number of permutations: 999
##
## Model: rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
## Df Variance F Pr(>F)
## RDA1 1 5.0745 20.672 0.001 ***
## Residual 48 11.7833
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# testing all axes sequentially (preceding axes are taken as constraints)
anova(rda,by="axis",model="direct",perm.max=9999,step=1000)
## Permutation test for rda under direct model
## Forward tests for axes
## Permutation: free
## Number of permutations: 999
##
## Model: rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
## Df Variance F Pr(>F)
## RDA1 1 5.0745 20.6715 0.001 ***
## RDA2 1 0.9454 3.8511 0.021 *
## RDA3 1 0.1968 0.8017 0.546
## Residual 48 11.7833
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# testing the individual terms=constraints
anova(rda,by="terms",model="direct",perm.max=9999,step=1000) # tests terms sequentially, order matters!
## Permutation test for rda under direct model
## Terms added sequentially (first to last)
## Permutation: free
## Number of permutations: 999
##
## Model: rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
## Df Variance F Pr(>F)
## logQ 1 1.8545 7.5544 0.002 **
## logTDN 1 3.7199 15.1532 0.001 ***
## canopy 1 0.6424 2.6168 0.035 *
## Residual 48 11.7833
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(rda,by="margin",model="direct",perm.max=9999,step=1000) # tests each term in full model (like drop1() function)
## Permutation test for rda under direct model
## Marginal effects of terms
## Permutation: free
## Number of permutations: 999
##
## Model: rda(formula = zwc ~ logQ + logTDN + canopy, data = xmat)
## Df Variance F Pr(>F)
## logQ 1 1.1757 4.7895 0.001 ***
## logTDN 1 2.7447 11.1808 0.001 ***
## canopy 1 0.6424 2.6168 0.030 *
## Residual 48 11.7833
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
###################
# making triplots #
# again various types of scaling for the plotting step:
# scaling 1 "distance triplot"
# only angles between constraints and responses reflect their correlations (not angles among responses)
# distances among sites reflect their Euclidean distances
# scaling 2 "correlation triplot"
# all angles between constraints and responses reflect correlations
# distances among sites do not reflect their Euclidean distances
# in both scaling types sites can be projected on constraints and on responses
# factor constraints are shown as centroids instead of arrows, projecting works identical
# scaling 3 is compromise
# build an RDA scaling type 1 triplot
plot(rda,scaling=1)
(sites<-scores(rda,choices=c(1,2),display="sites",scaling=1))
## RDA1 RDA2
## 1 -0.47708706 0.757822982
## 2 -0.30183628 0.926881198
## 3 -0.19105498 0.696540192
## 4 -0.41224315 0.505887724
## 7 0.46909247 -0.080931541
## 8 1.00641502 0.171753365
## 9 -0.27611373 0.085501680
## 10 -0.33935950 0.565775956
## 11 0.24587596 0.293021499
## 12 0.76819362 -0.267838709
## 13 -0.36773357 0.018719325
## 14 0.74970240 -0.533469771
## 15 -0.65396251 -0.192396774
## 16 -0.67274089 0.249392401
## 17 -0.84008293 0.168166947
## 18 -0.43167242 0.256537512
## 19 -0.35386382 -0.042012613
## 20 -0.10870767 -0.108800496
## 21 -0.49679038 -0.126272434
## 22 -0.29965548 -0.160919100
## 23 -0.44147805 0.051005608
## 24 -0.78893823 -0.051391214
## 25 -0.20088493 -0.272602952
## 26 -0.06475602 -0.331860547
## 27 -0.13375163 -0.247246487
## 28 -0.18250376 -0.284494012
## 29 -0.14541984 -0.286976426
## 30 -0.38753906 -0.135994789
## 31 0.47932829 -0.005588143
## 32 0.28557445 0.071067374
## 33 0.01525870 -0.393369015
## 34 0.05731908 0.312456891
## 35 -0.12847735 -0.151924742
## 36 0.35627838 0.148430402
## 37 0.53137430 -0.151970659
## 38 0.40955951 -0.065345506
## 39 0.54857945 0.046989184
## 40 0.41011812 -0.014011206
## 41 0.73669169 -0.441643953
## 42 0.79405601 -0.298230203
## 43 0.37902353 0.095862731
## 44 0.56395239 -0.289094360
## 45 0.67427788 0.528888976
## 46 -0.50063077 -0.013640487
## 47 -0.14028494 0.053125379
## 48 0.27970035 -0.027469107
## 49 -0.12008689 0.125297375
## 50 -0.29247055 -0.157474788
## 51 -0.28998363 -0.077414833
## 52 0.38467865 -0.355214393
## 53 -0.02719593 -0.221475935
## 54 -0.07774435 -0.342049506
## attr(,"const")
## [1] 5.504409
(lcs<-scores(rda,choices=c(1,2),display="lc",scaling=1)) # fitted/constrained site scores
## RDA1 RDA2
## 1 -0.104224288 0.189567711
## 2 0.242615728 0.276213147
## 3 -0.082327802 0.214791744
## 4 -0.185909806 -0.034824199
## 7 0.342526429 -0.167589611
## 8 0.743324762 -0.021603400
## 9 -0.503905543 -0.173366219
## 10 0.212860509 0.268209766
## 11 0.362683168 -0.041897420
## 12 0.751476304 -0.284475586
## 13 -0.477950182 0.304419950
## 14 0.042478939 0.351977893
## 15 -0.626142621 -0.006955898
## 16 -0.548623320 0.118470198
## 17 -0.595401031 -0.170611798
## 18 -0.351795708 0.172146937
## 19 -0.488744614 0.104907983
## 20 -0.311312613 0.046373049
## 21 -0.483994623 -0.136266840
## 22 -0.402846352 -0.118087635
## 23 -0.492790671 0.047052871
## 24 -0.479575026 -0.103167648
## 25 -0.195399738 -0.113840754
## 26 -0.047705540 -0.073898965
## 27 -0.370466946 0.039799254
## 28 -0.183817338 -0.077103367
## 29 -0.139722215 -0.133318827
## 30 -0.420364221 -0.014031283
## 31 0.490214711 0.055929670
## 32 0.262159763 0.061961130
## 33 -0.121128764 -0.302703414
## 34 0.172723906 0.408727984
## 35 -0.186562979 -0.164991565
## 36 0.433357781 0.021583490
## 37 0.298459410 0.221966799
## 38 0.445138750 -0.000400045
## 39 0.545922368 0.074661431
## 40 0.439088339 -0.017917235
## 41 0.776737283 -0.211610578
## 42 0.700910966 -0.243588247
## 43 0.504223931 0.190216605
## 44 0.556287538 -0.146420841
## 45 0.256778620 0.358787439
## 46 -0.281224871 -0.257363258
## 47 -0.004376186 -0.075815134
## 48 0.223422164 0.121212240
## 49 0.050314794 0.040156741
## 50 -0.471308404 -0.049664526
## 51 -0.458962873 -0.020184905
## 52 0.311539183 -0.225348633
## 53 -0.072579521 -0.142264087
## 54 -0.076081553 -0.159822114
## attr(,"const")
## [1] 5.504409
(species<-scores(rda,choices=c(1,2),display="sp",scaling=1)*0.5)
## RDA1 RDA2
## cond -0.54292631 1.335388807
## pH 0.42828677 -0.138350484
## turb -0.78277590 0.003432039
## Alk 0.03279901 1.016618867
## TSS -0.66578993 0.479223111
## PO4 -0.29631964 -0.098546059
## NH4 -0.50722452 0.635572107
## NO2 -0.75884500 -0.422261198
## NO3 -1.08762377 -1.025380733
## Br -0.87032206 0.323438441
## Cl -0.74785116 0.004828477
## Fl -0.67905469 0.317798293
## SO4 -0.46297404 0.032605655
## Na -0.66140095 0.192196915
## K -0.30936605 0.285318316
## Ca -0.18232269 1.094299339
## Mg -0.22414502 0.740302771
## TDN -1.15180016 -0.891678553
## attr(,"const")
## [1] 5.504409
(constraints<-scores(rda,choices=c(1,2),display="bp",scaling=1)*2)
## RDA1 RDA2
## logQ 0.5477956 -0.31771737
## logTDN -1.0011204 -0.14438929
## canopy 0.5210455 -0.06541799
## attr(,"const")
## [1] 5.504409
plot(sites,asp=1,pch=21,bg=landuse,ylim=c(-1.5,1.5))
Arrows(x0=0,y0=0,x1=constraints[,1],y1=constraints[,2],lwd=1.5,col="blue")
text(constraints[,1:2]*1.1,label=rownames(constraints),pos=4,cex=0.8,col="blue")
Arrows(x0=0,y0=0,x1=species[,1],y1=species[,2],lwd=1,arr.length=0)
text(species[,1:2]*1.1,label=rownames(species),pos=4,cex=0.6)
Two involved matrices: one dependent, one independent.
In the reciprocal averaging of CA a constraint is included:
The result are axes which inform about species-site relationships, but which also have maximized correlation with linear combinations of (environmental) predictors.
Site scores:
data(varespec) # a R dataset on vegetation
data(varechem) # soil chemistry
head(varespec)
## Callvulg Empenigr Rhodtome Vaccmyrt Vaccviti Pinusylv Descflex Betupube
## 18 0.55 11.13 0.00 0.00 17.80 0.07 0.00 0
## 15 0.67 0.17 0.00 0.35 12.13 0.12 0.00 0
## 24 0.10 1.55 0.00 0.00 13.47 0.25 0.00 0
## 27 0.00 15.13 2.42 5.92 15.97 0.00 3.70 0
## 23 0.00 12.68 0.00 0.00 23.73 0.03 0.00 0
## 19 0.00 8.92 0.00 2.42 10.28 0.12 0.02 0
## Vacculig Diphcomp Dicrsp Dicrfusc Dicrpoly Hylosple Pleuschr Polypili
## 18 1.60 2.07 0.00 1.62 0.00 0.0 4.67 0.02
## 15 0.00 0.00 0.33 10.92 0.02 0.0 37.75 0.02
## 24 0.00 0.00 23.43 0.00 1.68 0.0 32.92 0.00
## 27 1.12 0.00 0.00 3.63 0.00 6.7 58.07 0.00
## 23 0.00 0.00 0.00 3.42 0.02 0.0 19.42 0.02
## 19 0.00 0.00 0.00 0.32 0.02 0.0 21.03 0.02
## Polyjuni Polycomm Pohlnuta Ptilcili Barbhatc Cladarbu Cladrang Cladstel
## 18 0.13 0.00 0.13 0.12 0.00 21.73 21.47 3.50
## 15 0.23 0.00 0.03 0.02 0.00 12.05 8.13 0.18
## 24 0.23 0.00 0.32 0.03 0.00 3.58 5.52 0.07
## 27 0.00 0.13 0.02 0.08 0.08 1.42 7.63 2.55
## 23 2.12 0.00 0.17 1.80 0.02 9.08 9.22 0.05
## 19 1.58 0.18 0.07 0.27 0.02 7.23 4.95 22.08
## Cladunci Cladcocc Cladcorn Cladgrac Cladfimb Cladcris Cladchlo Cladbotr
## 18 0.30 0.18 0.23 0.25 0.25 0.23 0.00 0.00
## 15 2.65 0.13 0.18 0.23 0.25 1.23 0.00 0.00
## 24 8.93 0.00 0.20 0.48 0.00 0.07 0.10 0.02
## 27 0.15 0.00 0.38 0.12 0.10 0.03 0.00 0.02
## 23 0.73 0.08 1.42 0.50 0.17 1.78 0.05 0.05
## 19 0.25 0.10 0.25 0.18 0.10 0.12 0.05 0.02
## Cladamau Cladsp Cetreric Cetrisla Flavniva Nepharct Stersp Peltapht Icmaeric
## 18 0.08 0.02 0.02 0.00 0.12 0.02 0.62 0.02 0
## 15 0.00 0.00 0.15 0.03 0.00 0.00 0.85 0.00 0
## 24 0.00 0.00 0.78 0.12 0.00 0.00 0.03 0.00 0
## 27 0.00 0.02 0.00 0.00 0.00 0.00 0.00 0.07 0
## 23 0.00 0.00 0.00 0.00 0.02 0.00 1.58 0.33 0
## 19 0.00 0.00 0.00 0.00 0.02 0.00 0.28 0.00 0
## Cladcerv Claddefo Cladphyl
## 18 0 0.25 0
## 15 0 1.00 0
## 24 0 0.33 0
## 27 0 0.15 0
## 23 0 1.97 0
## 19 0 0.37 0
head(varechem)
## N P K Ca Mg S Al Fe Mn Zn Mo Baresoil Humdepth
## 18 19.8 42.1 139.9 519.4 90.0 32.3 39.0 40.9 58.1 4.5 0.3 43.9 2.2
## 15 13.4 39.1 167.3 356.7 70.7 35.2 88.1 39.0 52.4 5.4 0.3 23.6 2.2
## 24 20.2 67.7 207.1 973.3 209.1 58.1 138.0 35.4 32.1 16.8 0.8 21.2 2.0
## 27 20.6 60.8 233.7 834.0 127.2 40.7 15.4 4.4 132.0 10.7 0.2 18.7 2.9
## 23 23.8 54.5 180.6 777.0 125.8 39.5 24.2 3.0 50.1 6.6 0.3 46.0 3.0
## 19 22.8 40.9 171.4 691.8 151.4 40.8 104.8 17.6 43.6 9.1 0.4 40.5 3.8
## pH
## 18 2.7
## 15 2.8
## 24 3.0
## 27 2.8
## 23 2.7
## 19 2.7
apply(varespec,1,sum) # approximate 100 (total cover), "absolute" abundance data
## 18 15 24 27 23 19 22 16 28 13 14
## 89.20 89.82 94.21 125.61 90.46 81.27 109.76 88.52 110.70 101.89 81.65
## 20 25 7 5 6 3 4 2 9 12 10
## 64.11 94.06 103.38 94.77 110.90 106.67 84.83 119.13 122.60 119.80 122.37
## 11 21
## 112.84 99.17
# correspondence analysis #
# run a CA just based on the species data (unconstrained!)
vare.ca<-cca(X=varespec) # function also used for CCA, but here only one matrix X is supplied
summary(vare.ca,scaling=1)
##
## Call:
## cca(X = varespec)
##
## Partitioning of scaled Chi-square:
## Inertia Proportion
## Total 2.083 1
## Unconstrained 2.083 1
##
## Eigenvalues, and their contribution to the scaled Chi-square
##
## Importance of components:
## CA1 CA2 CA3 CA4 CA5 CA6 CA7
## Eigenvalue 0.5249 0.3568 0.2344 0.19546 0.17762 0.12156 0.11549
## Proportion Explained 0.2520 0.1713 0.1125 0.09383 0.08526 0.05835 0.05544
## Cumulative Proportion 0.2520 0.4233 0.5358 0.62962 0.71489 0.77324 0.82868
## CA8 CA9 CA10 CA11 CA12 CA13 CA14
## Eigenvalue 0.08894 0.07318 0.05752 0.04434 0.02546 0.01710 0.014896
## Proportion Explained 0.04269 0.03513 0.02761 0.02129 0.01222 0.00821 0.007151
## Cumulative Proportion 0.87137 0.90650 0.93411 0.95539 0.96762 0.97583 0.982978
## CA15 CA16 CA17 CA18 CA19 CA20
## Eigenvalue 0.010160 0.007830 0.006032 0.004008 0.002865 0.0019275
## Proportion Explained 0.004877 0.003759 0.002896 0.001924 0.001375 0.0009253
## Cumulative Proportion 0.987855 0.991614 0.994510 0.996434 0.997809 0.9987341
## CA21 CA22 CA23
## Eigenvalue 0.0018074 0.0005864 0.0002434
## Proportion Explained 0.0008676 0.0002815 0.0001168
## Cumulative Proportion 0.9996017 0.9998832 1.0000000
##
## Scaling 1 for species and site scores
## * Sites are scaled proportional to eigenvalues
## * Species are unscaled: weighted dispersion equal on all dimensions
##
##
## Species scores
##
## CA1 CA2 CA3 CA4 CA5 CA6
## Callvulg 0.0303167 -1.597460 0.11455 -2.894569 0.1376073 2.291129
## Empenigr 0.0751030 0.379305 0.39303 0.023675 0.8568729 -0.400964
## Rhodtome 1.1052309 1.499299 3.04284 0.120106 3.2324306 -0.283510
## Vaccmyrt 1.4614812 1.622935 2.72375 0.231688 0.4604556 0.712538
## Vaccviti 0.1468014 0.313436 0.14696 0.243505 0.6868371 -0.147815
## Pinusylv -0.4820096 0.588517 -0.36020 -0.127094 0.4064754 0.386604
## Descflex 1.5348239 1.218806 1.87562 -0.001340 -1.3136979 -0.070731
## Betupube 0.6694503 1.951826 3.84017 1.389423 7.5959115 -0.244478
## Vacculig -0.0830789 -1.629259 1.05063 0.802648 -0.3058811 -1.625341
## Diphcomp -0.5446464 -1.037570 0.52282 0.940275 0.3682126 -1.082929
## Dicrsp 1.8120408 0.360290 -4.92082 3.088562 1.3867372 0.157815
## Dicrfusc 1.2704743 -0.562978 -0.39718 -2.929542 0.3848272 -2.408710
## Dicrpoly 0.7248118 1.409347 0.80341 1.915549 4.5674148 1.295447
## Hylosple 2.0062408 1.743883 2.27549 0.928884 -3.7648428 2.254851
## Pleuschr 1.3102086 0.583036 -0.01004 0.137298 -1.1216144 0.200422
## Polypili -0.3805097 -1.243904 0.54593 1.477188 -0.7276341 -0.387641
## Polyjuni 1.0133795 0.099043 -2.24697 1.510641 0.7729714 -3.062378
## Polycomm 0.8468241 1.321773 1.13585 1.140723 2.6836594 -0.605038
## Pohlnuta -0.0136453 0.589290 -0.35542 0.135481 0.9369707 0.397246
## Ptilcili 0.4223631 1.598584 3.43474 1.400065 6.3209491 0.198935
## Barbhatc 0.5018348 2.119334 4.57303 1.693188 8.1101807 0.645995
## Cladarbu -0.1531729 -1.483884 0.20024 0.193680 0.0734141 0.358926
## Cladrang -0.5502561 -1.084008 0.40552 0.724060 -0.3357992 -0.335924
## Cladstel -1.4373146 1.077753 -0.44397 -0.375926 -0.2421525 0.004212
## Cladunci 0.8151727 -1.006186 -1.82587 -1.389523 1.6046713 3.675908
## Cladcocc -0.2133215 -0.584429 -0.21434 -0.567886 -0.0003788 -0.145303
## Cladcorn 0.2631227 -0.177858 -0.44464 0.272422 0.3992282 -0.306738
## Cladgrac 0.1956947 -0.311167 -0.23894 0.379013 0.4933026 0.037581
## Cladfimb 0.0009213 -0.161418 0.18463 -0.435908 0.4831233 -0.143751
## Cladcris 0.3373031 -0.470369 -0.05093 -0.823855 0.7182250 0.636140
## Cladchlo -0.6200021 1.207278 0.21889 0.426447 1.9506082 0.120722
## Cladbotr 0.5647242 1.047333 2.65330 0.907734 4.4946805 1.201655
## Cladamau -0.6598144 -1.512880 0.83251 1.577699 -0.0407227 -1.419139
## Cladsp -0.8209003 0.476164 -0.49752 -0.998241 -0.2393208 0.390785
## Cetreric 0.2458192 -0.689228 -1.68427 -0.131681 0.7439412 2.374535
## Cetrisla -0.3465221 1.362693 0.85897 0.396752 2.7526968 0.396591
## Flavniva -1.4391907 -0.833589 -0.12919 0.007071 -1.4841375 2.956977
## Nepharct 1.6813309 0.199484 -4.33509 2.229917 0.9561223 -5.472858
## Stersp -0.5172793 -2.280900 0.99775 2.377013 -0.8892757 -1.441228
## Peltapht 0.4035858 -0.043265 0.04538 0.711040 0.1824679 -0.841227
## Icmaeric 0.0378754 -2.419595 0.72135 0.361302 -0.3736424 -2.092136
## Cladcerv -0.9232858 -0.005233 -1.22058 0.305290 -0.8142627 0.414135
## Claddefo 0.5190399 -0.496632 -0.15271 -0.695927 0.9042143 0.909191
## Cladphyl -1.2836161 1.155872 -0.79912 -0.741170 -0.1608002 0.490526
##
##
## Site scores (weighted averages of species scores)
##
## CA1 CA2 CA3 CA4 CA5 CA6
## 18 -0.108122 -0.53705 0.229574 0.24412 0.1405624 -0.14253
## 15 0.697118 -0.14441 -0.031788 -0.21743 -0.2738522 -0.08146
## 24 0.987603 0.15042 -1.348447 0.80472 0.3095168 0.46773
## 27 0.851765 0.49901 0.443559 0.12277 -0.4814871 0.07589
## 23 0.359881 -0.05608 0.145813 0.15087 0.2405263 -0.17770
## 19 0.003545 0.37017 0.027760 0.06168 -0.1158930 -0.03413
## 22 0.860732 -0.11504 0.110869 -1.02169 0.0772348 -0.60530
## 16 0.636936 -0.33250 0.001120 -0.79797 0.0130769 -0.54049
## 28 1.279352 0.81557 0.670053 0.23137 -0.8929976 0.41783
## 13 -0.195009 -0.80564 0.117686 -0.58286 -0.0007212 0.53071
## 14 0.528532 -0.70420 -0.517771 -0.86836 0.5713441 0.91671
## 20 0.382866 -0.18686 -0.004789 0.10156 0.0458125 0.21087
## 25 0.990715 0.11967 -1.110040 0.44929 0.1885902 -0.70694
## 7 -0.264704 -1.06013 0.334900 0.45973 -0.0326631 -0.19945
## 5 -0.428410 -1.20765 0.374344 0.74970 -0.2596294 -0.30467
## 6 -0.330534 -0.77498 0.130760 0.22391 0.0632686 0.09060
## 3 -0.899601 0.12075 -0.075742 0.03842 -0.1489585 -0.12031
## 4 -0.770294 -0.35351 -0.033779 -0.01795 -0.3007839 0.44303
## 2 -0.992193 0.50319 -0.157505 -0.07070 -0.1065172 -0.09928
## 9 -0.937173 0.78688 -0.258119 -0.19377 -0.0343535 -0.01259
## 12 -0.726413 0.49163 -0.157235 -0.08698 -0.0105774 -0.02801
## 10 -1.002083 0.71239 -0.236526 -0.18643 -0.0231666 -0.04928
## 11 -0.322647 -0.03871 -0.001297 0.09029 -0.1481448 0.06934
## 21 0.259527 0.80746 1.124258 0.36083 1.5437866 0.07051
# summary(vare.ca,scaling=2)
# again two different types of scaling are possible for biplots
# scaling 1 (distances among sites matter)
# distances among sites approximate their chi^2 distance
# close sites have similar species abundances
# a site, which is near a specific species, has a high contribution of that species
# scaling 2 (relationships among species matter)
# distances among species approximate their chi^2 distance
# close species have similar abundances across sites
# a species, which is near a specific site, is more likely to be found at that site
plot(vare.ca,scaling=1)
# for any scaling take care when interpreting species close to origin:
# these are "everywhere" or have optimum right at the origin (i.e., optimum with regard to both axis shown)
# for more controlled plotting
species.scores<-scores(vare.ca,display="species",scaling=2)
site.scores<-scores(vare.ca,display="sites",scaling=2)
plot(site.scores,col="black",pch=21,xlim=c(-2,2),ylim=c(-2,2))
text(species.scores,col="red",label=names(varespec),cex=0.7)
# post-hoc fitting of an environmental variable
names(varechem)
## [1] "N" "P" "K" "Ca" "Mg" "S"
## [7] "Al" "Fe" "Mn" "Zn" "Mo" "Baresoil"
## [13] "Humdepth" "pH"
(ef<-envfit(vare.ca,varechem[,12:13],permutations=1999))
##
## ***VECTORS
##
## CA1 CA2 r2 Pr(>r)
## Baresoil 0.97947 -0.20161 0.2533 0.0505 .
## Humdepth 0.91602 0.40112 0.4524 0.0020 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## Permutation: free
## Number of permutations: 1999
plot(ef)
###########################
# canonical correspondence analysis #
vare.cca<-cca(Y=varespec,X=varechem) # note strange terminology of X and Y in vegan (don´t ask)
vare.cca<-cca(varespec~.,varechem) # hypothesis tests need formula interface (don´t ask)
summary(vare.cca,scaling=1)
##
## Call:
## cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
##
## Partitioning of scaled Chi-square:
## Inertia Proportion
## Total 2.0832 1.000
## Constrained 1.4415 0.692
## Unconstrained 0.6417 0.308
##
## Eigenvalues, and their contribution to the scaled Chi-square
##
## Importance of components:
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6 CCA7
## Eigenvalue 0.4389 0.2918 0.16285 0.14213 0.11795 0.08903 0.07029
## Proportion Explained 0.2107 0.1401 0.07817 0.06823 0.05662 0.04274 0.03374
## Cumulative Proportion 0.2107 0.3507 0.42890 0.49713 0.55375 0.59649 0.63023
## CCA8 CCA9 CCA10 CCA11 CCA12 CCA13
## Eigenvalue 0.05836 0.03114 0.013294 0.008364 0.006538 0.006156
## Proportion Explained 0.02801 0.01495 0.006382 0.004015 0.003139 0.002955
## Cumulative Proportion 0.65825 0.67319 0.679576 0.683592 0.686730 0.689685
## CCA14 CA1 CA2 CA3 CA4 CA5 CA6
## Eigenvalue 0.004733 0.19776 0.14193 0.10117 0.07079 0.05330 0.03330
## Proportion Explained 0.002272 0.09493 0.06813 0.04857 0.03398 0.02559 0.01598
## Cumulative Proportion 0.691958 0.78689 0.85502 0.90359 0.93757 0.96315 0.97914
## CA7 CA8 CA9
## Eigenvalue 0.018868 0.015104 0.009488
## Proportion Explained 0.009057 0.007251 0.004554
## Cumulative Proportion 0.988195 0.995446 1.000000
##
## Accumulated constrained eigenvalues
## Importance of components:
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6 CCA7
## Eigenvalue 0.4389 0.2918 0.1628 0.1421 0.11795 0.08903 0.07029
## Proportion Explained 0.3045 0.2024 0.1130 0.0986 0.08183 0.06176 0.04877
## Cumulative Proportion 0.3045 0.5069 0.6198 0.7184 0.80027 0.86203 0.91080
## CCA8 CCA9 CCA10 CCA11 CCA12 CCA13
## Eigenvalue 0.05836 0.03114 0.013294 0.008364 0.006538 0.006156
## Proportion Explained 0.04049 0.02160 0.009223 0.005803 0.004536 0.004271
## Cumulative Proportion 0.95128 0.97288 0.982107 0.987910 0.992446 0.996716
## CCA14
## Eigenvalue 0.004733
## Proportion Explained 0.003284
## Cumulative Proportion 1.000000
##
## Scaling 1 for species and site scores
## * Sites are scaled proportional to eigenvalues
## * Species are unscaled: weighted dispersion equal on all dimensions
##
##
## Species scores
##
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6
## Callvulg 0.11374 -1.73246 4.15753 1.844837 3.13741 -1.15626
## Empenigr -0.27373 0.14089 0.09036 -1.134550 -0.40226 0.03525
## Rhodtome -1.59033 -0.11156 0.19187 -2.490433 -0.62292 -1.73616
## Vaccmyrt -1.92827 0.56943 0.75259 -0.244263 -1.65624 -2.05452
## Vaccviti -0.23029 0.22315 -0.13141 -0.960947 0.24441 0.02995
## Pinusylv 0.36674 0.48934 0.55326 -0.726273 0.85051 -0.21227
## Descflex -2.17952 0.50020 -0.40165 1.608949 -1.38617 1.28224
## Betupube -1.07326 -0.41989 -0.20569 -6.388345 -0.62954 -5.60316
## Vacculig 0.77560 -2.19992 -0.93608 0.469588 -2.78966 1.04277
## Diphcomp 0.14991 -1.65300 -1.03898 -1.412059 -0.78833 2.08551
## Dicrsp -1.28302 0.42862 -4.34136 0.691801 4.43282 1.30777
## Dicrfusc -0.75393 -0.76901 2.04376 -0.684761 0.32655 2.14058
## Dicrpoly -0.79564 0.14903 -2.01239 -3.186680 2.23820 -3.43647
## Hylosple -2.75940 1.46965 0.12345 3.602353 -2.66866 -0.74851
## Pleuschr -1.39625 0.62360 -0.02266 0.817213 -0.19077 0.06281
## Polypili 0.21763 -0.84392 -1.27708 -0.747467 -0.15333 0.16978
## Polyjuni -0.91607 0.38917 -0.87255 -0.891253 -1.78446 1.17847
## Polycomm -1.34974 0.59359 -0.58214 -2.854381 -1.19037 -2.60320
## Pohlnuta -0.01435 0.46778 -0.34834 -0.931562 1.23465 -0.32446
## Ptilcili -0.86964 -0.22650 -0.14520 -5.594842 -0.48392 -5.05263
## Barbhatc -1.04773 -0.42524 -0.29330 -6.830157 -0.50320 -6.88497
## Cladarbu 0.31928 -1.31813 -0.06534 0.138503 -0.11811 -0.26229
## Cladrang 0.57516 -1.14185 -0.60438 0.280955 -0.47617 0.10938
## Cladstel 1.36834 1.29986 0.20555 0.179762 -0.04827 0.09185
## Cladunci -0.34820 0.11797 -0.03422 -1.037583 2.65119 -0.48962
## Cladcocc 0.33121 -0.25213 0.31806 -0.205437 0.09828 0.41903
## Cladcorn -0.34025 0.12974 -0.22432 -0.686052 -0.31883 0.57211
## Cladgrac -0.16429 -0.34432 -0.39566 -0.533216 0.70217 -0.07237
## Cladfimb 0.03022 -0.16993 0.47734 -0.696051 -0.10470 -0.11656
## Cladcris -0.20689 0.02979 1.04812 -1.124294 0.40186 -0.43505
## Cladchlo 0.66964 1.02386 -0.68975 -1.528620 0.49217 -0.75368
## Cladbotr -1.02718 -0.35199 0.48348 -3.528217 0.63524 -4.23041
## Cladamau -0.02415 -2.15363 -1.80591 -1.323302 -1.02050 2.39498
## Cladsp 1.03577 0.72454 0.76099 0.741439 1.75911 0.41843
## Cetreric 0.09754 -0.07199 -1.05941 0.315233 2.75328 -0.58261
## Cetrisla 0.24027 0.64936 -0.12182 -2.346145 0.48511 -2.31098
## Flavniva 1.31684 -1.19677 -1.15320 5.202080 1.07346 -7.81575
## Nepharct -1.15139 0.36798 -1.38414 -0.153781 -3.31081 2.49381
## Stersp 0.18370 -2.37391 -2.38790 -0.009846 -1.07525 1.39790
## Peltapht -0.60047 0.31181 0.12300 -0.899163 -0.76856 0.65021
## Icmaeric 0.26085 -2.83828 -1.06550 -0.409685 -1.20472 1.06913
## Cladcerv 1.06877 -0.10889 -0.78377 3.250753 0.01418 -3.50019
## Claddefo -0.45498 -0.03870 0.60324 -1.497542 0.85219 -0.63272
## Cladphyl 1.51291 2.08492 0.04117 -0.268421 0.27480 0.48796
##
##
## Site scores (weighted averages of species scores)
##
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6
## 18 0.11823 -0.57251 -0.164982 -0.22892 -0.1940177 0.07213
## 15 -0.64276 -0.10649 0.169910 0.11432 0.0521025 0.23988
## 24 -0.84786 0.25736 -1.189184 0.14813 1.3580797 0.22853
## 27 -0.99432 0.35227 0.034639 0.28730 -0.4232940 -0.02911
## 23 -0.39622 -0.09941 -0.054725 -0.43892 -0.1038877 0.02098
## 19 -0.07306 0.38585 0.006695 -0.02930 -0.1896126 -0.02464
## 22 -0.72351 -0.26482 0.855780 -0.16216 0.0893297 0.55882
## 16 -0.50071 -0.42518 0.666714 -0.05991 0.1632133 0.51821
## 28 -1.48534 0.62159 0.100450 0.70953 -0.6209907 -0.35786
## 13 0.26728 -0.79352 0.904036 0.45978 0.6372512 -0.27314
## 14 -0.30232 -0.37451 0.439688 -0.39404 0.9278442 0.04663
## 20 -0.36984 -0.13664 -0.135727 -0.13735 0.0942856 0.03259
## 25 -0.85602 0.13551 -0.587776 -0.01017 0.3304823 0.65496
## 7 0.36936 -1.08951 -0.372699 0.05638 -0.4616061 0.05740
## 5 0.44060 -1.21454 -0.658393 0.16630 -0.4226872 0.15976
## 6 0.39218 -0.69770 -0.189710 -0.03141 -0.0990151 -0.05450
## 3 0.88634 0.21282 -0.085773 0.09809 -0.2111364 0.08974
## 4 0.77344 -0.30247 -0.083929 0.80863 0.1228687 -0.94716
## 2 0.93351 0.60859 0.004559 0.01574 -0.1379707 0.08149
## 9 0.86982 0.91296 0.096369 -0.05063 0.0005494 0.01469
## 12 0.67010 0.58561 0.034417 -0.09231 -0.0424654 0.05488
## 10 0.93439 0.83587 0.093851 -0.06296 -0.0540425 0.05003
## 11 0.30814 0.02923 -0.059108 0.09765 -0.0281526 -0.01160
## 21 -0.47641 0.23201 0.003915 -1.44448 -0.2880137 -1.21174
##
##
## Site constraints (linear combinations of constraining variables)
##
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6
## 18 -0.28028 -0.71553 -0.198602 -0.35622 -0.016645 0.28042
## 15 -0.12605 0.26839 0.183427 -0.19962 -0.026309 -0.23569
## 24 -0.57190 0.13619 -1.113919 0.21486 1.130853 0.07844
## 27 -1.12491 0.26289 -0.227400 0.40474 -0.210924 0.14882
## 23 -0.52704 0.05792 0.103917 -0.34088 -0.098759 0.13090
## 19 -0.44851 0.54086 0.013494 -0.37832 -0.048521 -0.27997
## 22 -0.54244 -0.36270 0.612068 -0.02209 0.194629 0.66118
## 16 -0.09855 -0.62779 0.413117 -0.16871 -0.053130 -0.07504
## 28 -1.37258 0.59298 0.200796 0.71143 -0.478758 -0.19023
## 13 0.10953 -0.73196 1.049988 0.47179 0.604495 -0.16296
## 14 -0.09320 0.10867 0.313802 -0.33147 0.232443 -0.11453
## 20 -0.45423 0.04379 -0.082409 -0.42046 0.381970 -0.22780
## 25 -0.59995 0.15944 -0.222688 -0.02782 -0.388701 0.24252
## 7 0.91721 -1.04185 -0.323016 0.13738 -0.567910 -0.03541
## 5 0.06432 -1.09164 -0.636766 0.01507 -0.151542 0.29546
## 6 0.27735 -0.30740 -0.130894 0.02489 -0.019959 0.10057
## 3 0.63365 0.06729 -0.206032 0.05714 -0.365798 -0.04820
## 4 0.56735 -0.42871 -0.189592 0.87651 0.160886 -0.84789
## 2 1.01789 0.50232 0.038999 0.09780 -0.003433 0.21274
## 9 1.01611 0.86649 -0.006136 -0.04395 0.239962 0.19821
## 12 0.29646 0.12958 0.378874 -0.10628 0.044242 0.11423
## 10 0.73605 0.86077 -0.016804 0.04149 -0.158371 0.07950
## 11 0.39119 0.19766 -0.018368 -0.05333 -0.024357 -0.11579
## 21 -0.45499 -0.12585 -0.070007 -1.04926 -0.070611 -0.65096
##
##
## Biplot scores for constraining variables
##
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6
## N -0.14766 -0.28570 0.002715 0.066860 -0.086965 0.0304388
## P -0.21110 0.31268 -0.065374 0.180759 0.063227 -0.0363527
## K -0.24254 0.16634 0.145204 0.180743 0.111771 -0.0586722
## Ca -0.29655 0.22782 -0.015240 0.037048 0.105769 0.0129929
## Mg -0.28817 0.18393 -0.057371 0.040679 0.170979 -0.0017181
## S -0.01594 0.22454 0.059879 0.167562 0.205056 -0.0496189
## Al 0.50996 -0.02564 0.015177 0.147400 0.055261 -0.1004202
## Fe 0.43000 -0.04760 -0.016976 0.099139 -0.023974 -0.0332227
## Mn -0.47852 0.12132 0.045621 0.109904 -0.047628 0.0538484
## Zn -0.23723 0.18092 -0.112151 0.130337 0.212656 -0.0003564
## Mo 0.13523 -0.05582 -0.063359 0.122240 0.177367 -0.0935487
## Baresoil -0.35558 -0.13762 0.055249 -0.196250 0.057225 -0.1051508
## Humdepth -0.46157 0.10891 0.109612 -0.051173 -0.001117 -0.0153218
## pH 0.32935 0.04056 -0.131693 0.007887 -0.049995 -0.0176316
summary(vare.cca,scaling=2)
##
## Call:
## cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
##
## Partitioning of scaled Chi-square:
## Inertia Proportion
## Total 2.0832 1.000
## Constrained 1.4415 0.692
## Unconstrained 0.6417 0.308
##
## Eigenvalues, and their contribution to the scaled Chi-square
##
## Importance of components:
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6 CCA7
## Eigenvalue 0.4389 0.2918 0.16285 0.14213 0.11795 0.08903 0.07029
## Proportion Explained 0.2107 0.1401 0.07817 0.06823 0.05662 0.04274 0.03374
## Cumulative Proportion 0.2107 0.3507 0.42890 0.49713 0.55375 0.59649 0.63023
## CCA8 CCA9 CCA10 CCA11 CCA12 CCA13
## Eigenvalue 0.05836 0.03114 0.013294 0.008364 0.006538 0.006156
## Proportion Explained 0.02801 0.01495 0.006382 0.004015 0.003139 0.002955
## Cumulative Proportion 0.65825 0.67319 0.679576 0.683592 0.686730 0.689685
## CCA14 CA1 CA2 CA3 CA4 CA5 CA6
## Eigenvalue 0.004733 0.19776 0.14193 0.10117 0.07079 0.05330 0.03330
## Proportion Explained 0.002272 0.09493 0.06813 0.04857 0.03398 0.02559 0.01598
## Cumulative Proportion 0.691958 0.78689 0.85502 0.90359 0.93757 0.96315 0.97914
## CA7 CA8 CA9
## Eigenvalue 0.018868 0.015104 0.009488
## Proportion Explained 0.009057 0.007251 0.004554
## Cumulative Proportion 0.988195 0.995446 1.000000
##
## Accumulated constrained eigenvalues
## Importance of components:
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6 CCA7
## Eigenvalue 0.4389 0.2918 0.1628 0.1421 0.11795 0.08903 0.07029
## Proportion Explained 0.3045 0.2024 0.1130 0.0986 0.08183 0.06176 0.04877
## Cumulative Proportion 0.3045 0.5069 0.6198 0.7184 0.80027 0.86203 0.91080
## CCA8 CCA9 CCA10 CCA11 CCA12 CCA13
## Eigenvalue 0.05836 0.03114 0.013294 0.008364 0.006538 0.006156
## Proportion Explained 0.04049 0.02160 0.009223 0.005803 0.004536 0.004271
## Cumulative Proportion 0.95128 0.97288 0.982107 0.987910 0.992446 0.996716
## CCA14
## Eigenvalue 0.004733
## Proportion Explained 0.003284
## Cumulative Proportion 1.000000
##
## Scaling 2 for species and site scores
## * Species are scaled proportional to eigenvalues
## * Sites are unscaled: weighted dispersion equal on all dimensions
##
##
## Species scores
##
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6
## Callvulg 0.075347 -0.93581 1.677742 0.695507 1.077518 -0.345001
## Empenigr -0.181340 0.07610 0.036462 -0.427727 -0.138153 0.010517
## Rhodtome -1.053549 -0.06026 0.077428 -0.938897 -0.213938 -0.518031
## Vaccmyrt -1.277428 0.30759 0.303704 -0.092088 -0.568820 -0.613023
## Vaccviti -0.152563 0.12054 -0.053031 -0.362279 0.083942 0.008938
## Pinusylv 0.242956 0.26432 0.223265 -0.273806 0.292102 -0.063335
## Descflex -1.443872 0.27019 -0.162082 0.606576 -0.476067 0.382590
## Betupube -0.711004 -0.22681 -0.083007 -2.408417 -0.216212 -1.671857
## Vacculig 0.513817 -1.18831 -0.377748 0.177035 -0.958084 0.311138
## Diphcomp 0.099310 -0.89289 -0.419273 -0.532348 -0.270745 0.622270
## Dicrsp -0.849964 0.23153 -1.751924 0.260810 1.522412 0.390210
## Dicrfusc -0.499460 -0.41539 0.824743 -0.258156 0.112149 0.638702
## Dicrpoly -0.527090 0.08050 -0.812083 -1.201383 0.768689 -1.025365
## Hylosple -1.828026 0.79385 0.049816 1.358093 -0.916528 -0.223338
## Pleuschr -0.924978 0.33684 -0.009146 0.308091 -0.065518 0.018741
## Polypili 0.144172 -0.45586 -0.515356 -0.281796 -0.052660 0.050659
## Polyjuni -0.606869 0.21021 -0.352109 -0.336004 -0.612858 0.351629
## Polycomm -0.894165 0.32063 -0.234919 -1.076106 -0.408823 -0.776736
## Pohlnuta -0.009508 0.25268 -0.140571 -0.351201 0.424031 -0.096811
## Ptilcili -0.576115 -0.12234 -0.058593 -2.109265 -0.166198 -1.507591
## Barbhatc -0.694092 -0.22970 -0.118360 -2.574980 -0.172821 -2.054320
## Cladarbu 0.211517 -0.71201 -0.026366 0.052216 -0.040564 -0.078262
## Cladrang 0.381030 -0.61678 -0.243893 0.105921 -0.163536 0.032637
## Cladstel 0.906486 0.70213 0.082949 0.067771 -0.016579 0.027407
## Cladunci -0.230671 0.06372 -0.013810 -0.391170 0.910527 -0.146092
## Cladcocc 0.219419 -0.13619 0.128350 -0.077450 0.033754 0.125028
## Cladcorn -0.225404 0.07008 -0.090524 -0.258643 -0.109501 0.170706
## Cladgrac -0.108836 -0.18599 -0.159664 -0.201023 0.241156 -0.021594
## Cladfimb 0.020022 -0.09179 0.192626 -0.262413 -0.035959 -0.034780
## Cladcris -0.137056 0.01609 0.422960 -0.423861 0.138016 -0.129810
## Cladchlo 0.443621 0.55305 -0.278345 -0.576292 0.169030 -0.224882
## Cladbotr -0.680481 -0.19013 0.195105 -1.330144 0.218169 -1.262258
## Cladamau -0.015996 -1.16331 -0.728763 -0.498887 -0.350481 0.714608
## Cladsp 0.686166 0.39137 0.307091 0.279524 0.604150 0.124850
## Cetreric 0.064619 -0.03889 -0.427516 0.118844 0.945590 -0.173838
## Cetrisla 0.159171 0.35076 -0.049161 -0.884501 0.166607 -0.689545
## Flavniva 0.872373 -0.64645 -0.465365 1.961193 0.368671 -2.332045
## Nepharct -0.762768 0.19877 -0.558560 -0.057976 -1.137069 0.744096
## Stersp 0.121697 -1.28229 -0.963619 -0.003712 -0.369284 0.417103
## Peltapht -0.397796 0.16843 0.049634 -0.338986 -0.263955 0.194009
## Icmaeric 0.172805 -1.53313 -0.429975 -0.154452 -0.413750 0.319003
## Cladcerv 0.708032 -0.05882 -0.316283 1.225539 0.004871 -1.044377
## Claddefo -0.301412 -0.02090 0.243431 -0.564576 0.292677 -0.188788
## Cladphyl 1.002262 1.12620 0.016613 -0.101195 0.094379 0.145598
##
##
## Site scores (weighted averages of species scores)
##
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6
## 18 0.1785 -1.05988 -0.408835 -0.60721 -0.56492 0.24175
## 15 -0.9702 -0.19714 0.421046 0.30324 0.15171 0.80394
## 24 -1.2798 0.47645 -2.946863 0.39292 3.95433 0.76592
## 27 -1.5009 0.65216 0.085837 0.76207 -1.23251 -0.09756
## 23 -0.5981 -0.18404 -0.135611 -1.16425 -0.30249 0.07033
## 19 -0.1103 0.71431 0.016591 -0.07773 -0.55210 -0.08258
## 22 -1.0921 -0.49026 2.120668 -0.43014 0.26010 1.87287
## 16 -0.7558 -0.78712 1.652152 -0.15892 0.47523 1.73677
## 28 -2.2421 1.15075 0.248921 1.88204 -1.80814 -1.19935
## 13 0.4035 -1.46904 2.240249 1.21956 1.85549 -0.91541
## 14 -0.4563 -0.69333 1.089571 -1.04519 2.70161 0.15628
## 20 -0.5583 -0.25296 -0.336340 -0.36433 0.27453 0.10923
## 25 -1.2922 0.25087 -1.456542 -0.02698 0.96227 2.19508
## 7 0.5576 -2.01700 -0.923568 0.14954 -1.34406 0.19237
## 5 0.6651 -2.24847 -1.631533 0.44110 -1.23074 0.53544
## 6 0.5920 -1.29165 -0.470112 -0.08331 -0.28830 -0.18265
## 3 1.3379 0.39399 -0.212551 0.26020 -0.61477 0.30075
## 4 1.1675 -0.55997 -0.207980 2.14490 0.35776 -3.17436
## 2 1.4091 1.12669 0.011297 0.04175 -0.40173 0.27311
## 9 1.3130 1.69016 0.238808 -0.13429 0.00160 0.04923
## 12 1.0115 1.08413 0.085287 -0.24485 -0.12365 0.18392
## 10 1.4105 1.54744 0.232569 -0.16699 -0.15736 0.16768
## 11 0.4651 0.05411 -0.146473 0.25902 -0.08197 -0.03886
## 21 -0.7191 0.42952 0.009702 -3.83149 -0.83861 -4.06109
##
##
## Site constraints (linear combinations of constraining variables)
##
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6
## 18 -0.42308 -1.32466 -0.49215 -0.94489 -0.048464 0.9398
## 15 -0.19026 0.49687 0.45454 -0.52951 -0.076603 -0.7899
## 24 -0.86328 0.25213 -2.76035 0.56993 3.292710 0.2629
## 27 -1.69805 0.48669 -0.56351 1.07358 -0.614147 0.4988
## 23 -0.79557 0.10723 0.25751 -0.90419 -0.287557 0.4387
## 19 -0.67702 1.00130 0.03344 -1.00351 -0.141279 -0.9383
## 22 -0.81881 -0.67147 1.51674 -0.05858 0.566703 2.2159
## 16 -0.14877 -1.16222 1.02373 -0.44751 -0.154699 -0.2515
## 28 -2.07190 1.09778 0.49758 1.88707 -1.394002 -0.6375
## 13 0.16534 -1.35508 2.60193 1.25142 1.760111 -0.5461
## 14 -0.14069 0.20118 0.77762 -0.87922 0.676806 -0.3838
## 20 -0.68566 0.08107 -0.20421 -1.11529 1.112185 -0.7635
## 25 -0.90562 0.29517 -0.55183 -0.07379 -1.131782 0.8128
## 7 1.38453 -1.92877 -0.80045 0.36440 -1.653585 -0.1187
## 5 0.09709 -2.02095 -1.57794 0.03999 -0.441247 0.9902
## 6 0.41866 -0.56908 -0.32436 0.06603 -0.058116 0.3371
## 3 0.95649 0.12458 -0.51056 0.15157 -1.065096 -0.1616
## 4 0.85641 -0.79366 -0.46982 2.32495 0.468453 -2.8417
## 2 1.53650 0.92994 0.09664 0.25941 -0.009995 0.7130
## 9 1.53381 1.60412 -0.01520 -0.11658 0.698700 0.6643
## 12 0.44751 0.23990 0.93887 -0.28191 0.128819 0.3828
## 10 1.11107 1.59354 -0.04164 0.11005 -0.461130 0.2664
## 11 0.59050 0.36592 -0.04552 -0.14145 -0.070919 -0.3881
## 21 -0.68681 -0.23299 -0.17348 -2.78317 -0.205599 -2.1817
##
##
## Biplot scores for constraining variables
##
## CCA1 CCA2 CCA3 CCA4 CCA5 CCA6
## N -0.22290 -0.52891 0.006729 0.17735 -0.253216 0.102014
## P -0.31866 0.57886 -0.162001 0.47947 0.184099 -0.121835
## K -0.36612 0.30794 0.359824 0.47942 0.325444 -0.196637
## Ca -0.44764 0.42176 -0.037765 0.09827 0.307969 0.043545
## Mg -0.43499 0.34051 -0.142169 0.10790 0.497841 -0.005758
## S -0.02406 0.41570 0.148384 0.44446 0.597063 -0.166296
## Al 0.76978 -0.04747 0.037610 0.39098 0.160905 -0.336554
## Fe 0.64909 -0.08811 -0.042067 0.26297 -0.069806 -0.111345
## Mn -0.72232 0.22460 0.113052 0.29152 -0.138680 0.180471
## Zn -0.35810 0.33493 -0.277916 0.34572 0.619191 -0.001195
## Mo 0.20413 -0.10334 -0.157007 0.32424 0.516439 -0.313525
## Baresoil -0.53675 -0.25477 0.136910 -0.52055 0.166621 -0.352409
## Humdepth -0.69673 0.20163 0.271625 -0.13574 -0.003252 -0.051350
## pH 0.49716 0.07509 -0.326341 0.02092 -0.145569 -0.059091
# again two different types of scaling are possible for triplots
# hypothesis tests #
# testing the first axis (global test)
anova(vare.cca)
## Permutation test for cca under reduced model
## Permutation: free
## Number of permutations: 999
##
## Model: cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
## Df ChiSquare F Pr(>F)
## Model 14 1.44148 1.4441 0.034 *
## Residual 9 0.64171
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(vare.cca,first=TRUE)
## Permutation test for cca under reduced model
## Permutation: free
## Number of permutations: 999
##
## Model: cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
## Df ChiSquare F Pr(>F)
## CCA1 1 0.43887 6.1551 0.027 *
## Residual 9 0.64171
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# testing all axes sequentially (preceding axes are taken as constraints)
anova(vare.cca,by="axis",model="direct",perm.max=9999,step=1000)
## Permutation test for cca under direct model
## Forward tests for axes
## Permutation: free
## Number of permutations: 999
##
## Model: cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
## Df ChiSquare F Pr(>F)
## CCA1 1 0.43887 6.1551 0.035 *
## CCA2 1 0.29178 4.0921 0.405
## CCA3 1 0.16285 2.2839 0.962
## CCA4 1 0.14213 1.9934 0.983
## CCA5 1 0.11795 1.6543 0.995
## CCA6 1 0.08903 1.2486 1.000
## CCA7 1 0.07029 0.9859 1.000
## CCA8 1 0.05836 0.8185 1.000
## CCA9 1 0.03114 0.4367 1.000
## CCA10 1 0.01329 0.1865 1.000
## CCA11 1 0.00836 0.1173 1.000
## CCA12 1 0.00654 0.0917 1.000
## CCA13 1 0.00616 0.0863 1.000
## CCA14 1 0.00473 0.0664 1.000
## Residual 9 0.64171
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# testing the individual terms=constraints
anova(vare.cca,by="terms",model="direct",perm.max=9999,step=1000) # tests terms sequentially, order matters!
## Permutation test for cca under direct model
## Terms added sequentially (first to last)
## Permutation: free
## Number of permutations: 999
##
## Model: cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
## Df ChiSquare F Pr(>F)
## N 1 0.13001 1.8234 0.066 .
## P 1 0.17996 2.5240 0.013 *
## K 1 0.13643 1.9134 0.077 .
## Ca 1 0.08614 1.2081 0.310
## Mg 1 0.06760 0.9481 0.482
## S 1 0.17545 2.4606 0.010 **
## Al 1 0.14797 2.0753 0.021 *
## Fe 1 0.04982 0.6987 0.720
## Mn 1 0.05080 0.7124 0.698
## Zn 1 0.07167 1.0052 0.425
## Mo 1 0.09585 1.3442 0.215
## Baresoil 1 0.08683 1.2178 0.283
## Humdepth 1 0.09502 1.3327 0.216
## pH 1 0.06794 0.9529 0.485
## Residual 9 0.64171
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
anova(vare.cca,by="margin",model="direct",perm.max=9999,step=1000) # tests each term in full model (like drop1() function)
## Permutation test for cca under direct model
## Marginal effects of terms
## Permutation: free
## Number of permutations: 999
##
## Model: cca(formula = varespec ~ N + P + K + Ca + Mg + S + Al + Fe + Mn + Zn + Mo + Baresoil + Humdepth + pH, data = varechem)
## Df ChiSquare F Pr(>F)
## N 1 0.06302 0.8838 0.545
## P 1 0.10327 1.4484 0.161
## K 1 0.10385 1.4565 0.117
## Ca 1 0.06169 0.8652 0.585
## Mg 1 0.08830 1.2384 0.257
## S 1 0.10388 1.4569 0.139
## Al 1 0.06876 0.9644 0.455
## Fe 1 0.04765 0.6683 0.777
## Mn 1 0.08110 1.1375 0.308
## Zn 1 0.06448 0.9043 0.550
## Mo 1 0.07761 1.0885 0.361
## Baresoil 1 0.08966 1.2574 0.280
## Humdepth 1 0.13548 1.9001 0.041 *
## pH 1 0.06794 0.9529 0.476
## Residual 9 0.64171
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
# quite a lot of variables in the constraining matrix, maybe selection would be adequate
# --> function ordistep()
###################
# making triplots #
# again various types of scaling for the plotting step:
# scaling 1 "distance triplot"
# sites can be projected on constraints
# sites close to centroid of factor constraint are more likely to possess the specific state (factor level)
# distances among sites reflect their Chi^2 distances
# scaling 2 "correlation triplot"
# species can be projected on constraints (to give their optimum)
# species close to centroid of factor constraint are more likely to be found in the respective sites
# distances among sites do not reflect their Chi^2 distances
# scaling 3 is compromise
plot(vare.cca,scaling=1,display=c("species","sites"))
plot(vare.cca,scaling=1,display=c("species","sites","bp"))
# for any scaling take care when interpreting species close to origin:
# these are "everywhere" or have optimum right at the origin (i.e., optimum with regard to both axis shown)
# for more controlled plotting compute scores individually
(species<-scores(vare.cca,display="species",scaling=2))
## CCA1 CCA2
## Callvulg 0.075346828 -0.93580864
## Empenigr -0.181339634 0.07610159
## Rhodtome -1.053549303 -0.06026078
## Vaccmyrt -1.277428382 0.30758571
## Vaccviti -0.152563159 0.12053851
## Pinusylv 0.242955730 0.26432438
## Descflex -1.443872268 0.27018775
## Betupube -0.711004196 -0.22680891
## Vacculig 0.513817276 -1.18831302
## Diphcomp 0.099309627 -0.89289046
## Dicrsp -0.849963650 0.23152562
## Dicrfusc -0.499459808 -0.41538963
## Dicrpoly -0.527089621 0.08050066
## Hylosple -1.828025570 0.79385044
## Pleuschr -0.924977860 0.33684271
## Polypili 0.144171903 -0.45585592
## Polyjuni -0.606869478 0.21021273
## Polycomm -0.894165298 0.32063347
## Pohlnuta -0.009508326 0.25267888
## Ptilcili -0.576115080 -0.12234488
## Barbhatc -0.694091803 -0.22969682
## Cladarbu 0.211516504 -0.71200604
## Cladrang 0.381029817 -0.61678185
## Cladstel 0.906485693 0.70213449
## Cladunci -0.230670950 0.06372124
## Cladcocc 0.219418695 -0.13619040
## Cladcorn -0.225403730 0.07008025
## Cladgrac -0.108836318 -0.18598627
## Cladfimb 0.020022402 -0.09178764
## Cladcris -0.137056217 0.01608948
## Cladchlo 0.443621232 0.55305019
## Cladbotr -0.680480562 -0.19013376
## Cladamau -0.015995645 -1.16331056
## Cladsp 0.686166357 0.39136838
## Cetreric 0.064619228 -0.03888872
## Cetrisla 0.159171117 0.35076227
## Flavniva 0.872372656 -0.64644751
## Nepharct -0.762767923 0.19876924
## Stersp 0.121696721 -1.28229477
## Peltapht -0.397796035 0.16842583
## Icmaeric 0.172805490 -1.53313439
## Cladcerv 0.708032101 -0.05881737
## Claddefo -0.301411810 -0.02090192
## Cladphyl 1.002261821 1.12619548
(lcs<-scores(vare.cca,display="lc",scaling=2)) # fitted site scores
## CCA1 CCA2
## 18 -0.42308410 -1.32465581
## 15 -0.19026462 0.49687345
## 24 -0.86328447 0.25212611
## 27 -1.69805066 0.48669129
## 23 -0.79556975 0.10723446
## 19 -0.67702233 1.00129602
## 22 -0.81881442 -0.67146524
## 16 -0.14876518 -1.16222054
## 28 -2.07190438 1.09778134
## 13 0.16533766 -1.35507570
## 14 -0.14068713 0.20117513
## 20 -0.68566132 0.08107181
## 25 -0.90561895 0.29516717
## 7 1.38453057 -1.92876582
## 5 0.09708596 -2.02094655
## 6 0.41865544 -0.56908384
## 3 0.95649196 0.12457945
## 4 0.85641341 -0.79366326
## 2 1.53649728 0.92993610
## 9 1.53380971 1.60412304
## 12 0.44750585 0.23989837
## 10 1.11106621 1.59354430
## 11 0.59049933 0.36592318
## 21 -0.68681020 -0.23299449
(sites<-scores(vare.cca,display="sites",scaling=2)) # unfitted site scores
## CCA1 CCA2
## 18 0.1784733 -1.05988423
## 15 -0.9702382 -0.19713866
## 24 -1.2798478 0.47644978
## 27 -1.5009195 0.65215591
## 23 -0.5980933 -0.18403623
## 19 -0.1102881 0.71431421
## 22 -1.0921288 -0.49026245
## 16 -0.7558244 -0.78712482
## 28 -2.2421137 1.15075172
## 13 0.4034539 -1.46904155
## 14 -0.4563466 -0.69332916
## 20 -0.5582740 -0.25295922
## 25 -1.2921584 0.25086578
## 7 0.5575516 -2.01700439
## 5 0.6650822 -2.24846553
## 6 0.5919915 -1.29165187
## 3 1.3379214 0.39399435
## 4 1.1674982 -0.55996655
## 2 1.4091299 1.12668910
## 9 1.3129824 1.69015916
## 12 1.0115068 1.08413112
## 10 1.4104517 1.54743675
## 11 0.4651334 0.05410696
## 21 -0.7191331 0.42951771
(constraints<-scores(vare.cca,choices=c(1,2),display="bp",scaling=2))
## CCA1 CCA2
## N -0.22289549 -0.52891359
## P -0.31865680 0.57886019
## K -0.36611630 0.30794063
## Ca -0.44763820 0.42176452
## Mg -0.43499067 0.34051494
## S -0.02405594 0.41569739
## Al 0.76977830 -0.04747443
## Fe 0.64908911 -0.08811470
## Mn -0.72232495 0.22459941
## Zn -0.35809796 0.33492792
## Mo 0.20413234 -0.10334347
## Baresoil -0.53674724 -0.25477013
## Humdepth -0.69673051 0.20163085
## pH 0.49715734 0.07508792
plot(sites,col="black",pch=21,xlim=c(-2,2),ylim=c(-2,2))
text(species,col="red",label=names(varespec),cex=0.7)
Arrows(x0=0,y0=0,x1=constraints[,1],y1=constraints[,2],lwd=1.5,col="blue")
text(constraints[,1:2]*1.1,label=rownames(constraints),pos=4,cex=0.8,col="blue")
Whenever compositional turnover at regional scale is of interest!
Diversity partitioning into \(\alpha\), \(\beta\), \(\gamma\):
\(\beta\)-diversity is linked to species-accumulation curves.
\[ D_{(a,b)}+D_{(b,c)}\geq{D_{(a,c)}} \] (think of a triangle: the distance from A to C via B must be larger than the direct distance from A to C)
\[ D_{(a,b)}+D_{(b,c)}<{D_{(a,c)}} \] … in which case perfect projections into Euclidean space are not possible. Two often used transformations to make semimetric coefficients analyzable in Euclidean space are adding a constant or \(\sqrt(D)\).
Hierarchical group-forming (clustering) based on pairwise (dis)similarity.
Three types:
Various “linkage rules” to group clusters in agglomerative clustering:
lipids<-read.table(file="data/BacterialMembrane.txt",header=TRUE)
names(lipids)
## [1] "case" "isolate" "temperature" "FA1_SAnb" "FA2_MU"
## [6] "FA3_SAnb" "FA4_SAb" "FA5_SAb" "FA6_SAnb" "FA7_MU"
## [11] "FA8_SAb" "FA9_SAnb" "FA10_MU" "FA11_SAb" "FA12_SAnb"
## [16] "FA13_MUb" "FA14_MU" "FA15_SAnb" "FA16_MU" "FA17_MU"
## [21] "sum_MU" "sum_SA" "sum_SAbran" "sum_SAnbran" "SA_branprop"
lip_data<-lipids[,4:20]
# abbreviations: MU = mono-unsaturated, SA = saturated, nb = non-branched, b = branched
# SA_branprop = proportion of branched and saturated FAs
# theory: unsaturated and branched FAs increase fluidity of membrane
# test (i) adaptation vs. (ii) acclimatization to temperature by changing FA composition of membranes
isolate<-factor(lipids$isolate)
temperature<-factor(lipids$temperature)
combifac<-factor(paste(temperature,"_",isolate,sep=""))
## distance matrix: Euclidean distance on arcsine-sqr-data
as_lip_data<-asin(sqrt(lip_data)) # very old-school, better don´t do ;-)
lipids_distE<-vegdist(as_lip_data, method="euclidean")
## alternatively: Bray-Curtis distance with proportional data
lipids_distBC<-vegdist(lip_data, method="bray")
# cluster analysis #
# various agglomeration methods available and the choice is important, explore method=
# "single": nearest neighbour counts (good for gradients, but makes chains)
# "complete": all group members must be close, farthest group member counts (makes small spheric groups, good to find outliers)
# "average": compromise average strategy, new member joins at mean distance to all group members, actually UPGMA
# "ward.D": aims at minimizing within-group sums of squares of distances
lipids_cluster<-hclust(lipids_distBC, method = "ward.D")
lipids_cluster$height<-sqrt(lipids_cluster$height) # may help
# then compare effect of agglomeration method on dendrogram
plot(lipids_cluster, hang = -1, labels = combifac, ylab = "BC")
# which agglomeration method (and thus dendrogram) is best?
cophenetic(lipids_cluster) # linkage distances in dendrogram
## 1 2 3 4 5 6 7
## 2 0.16296148
## 3 0.40441248 0.40441248
## 4 0.40441248 0.40441248 0.31840951
## 5 1.40874876 1.40874876 1.40874876 1.40874876
## 6 1.40874876 1.40874876 1.40874876 1.40874876 0.20777591
## 7 1.40874876 1.40874876 1.40874876 1.40874876 0.27324836 0.27324836
## 8 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876 1.40874876 1.40874876
## 9 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876 1.40874876 1.40874876
## 10 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876 1.40874876 1.40874876
## 11 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876 1.40874876 1.40874876
## 12 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876 1.40874876 1.40874876
## 13 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876 1.40874876 1.40874876
## 14 1.40874876 1.40874876 1.40874876 1.40874876 0.49516799 0.49516799 0.49516799
## 15 1.40874876 1.40874876 1.40874876 1.40874876 0.49516799 0.49516799 0.49516799
## 16 1.40874876 1.40874876 1.40874876 1.40874876 0.49516799 0.49516799 0.49516799
## 17 0.51155960 0.51155960 0.51155960 0.51155960 1.40874876 1.40874876 1.40874876
## 18 0.51155960 0.51155960 0.51155960 0.51155960 1.40874876 1.40874876 1.40874876
## 19 0.51155960 0.51155960 0.51155960 0.51155960 1.40874876 1.40874876 1.40874876
## 20 0.40441248 0.40441248 0.25690465 0.31840951 1.40874876 1.40874876 1.40874876
## 21 1.40874876 1.40874876 1.40874876 1.40874876 0.56164834 0.56164834 0.56164834
## 22 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876 1.40874876 1.40874876
## 23 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876 1.40874876 1.40874876
## 24 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876 1.40874876 1.40874876
## 25 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876 1.40874876 1.40874876
## 26 1.40874876 1.40874876 1.40874876 1.40874876 0.39214779 0.39214779 0.39214779
## 27 1.40874876 1.40874876 1.40874876 1.40874876 0.56164834 0.56164834 0.56164834
## 28 1.40874876 1.40874876 1.40874876 1.40874876 0.56164834 0.56164834 0.56164834
## 8 9 10 11 12 13 14
## 2
## 3
## 4
## 5
## 6
## 7
## 8
## 9 0.23904776
## 10 0.29578401 0.29578401
## 11 0.23904776 0.20171061 0.29578401
## 12 0.05572996 0.23904776 0.29578401 0.23904776
## 13 0.23904776 0.04176804 0.29578401 0.20171061 0.23904776
## 14 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876
## 15 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 0.16915116
## 16 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 0.24387639
## 17 0.97985304 0.97985304 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876
## 18 0.97985304 0.97985304 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876
## 19 0.97985304 0.97985304 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876
## 20 0.97985304 0.97985304 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876
## 21 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 0.56164834
## 22 0.49724975 0.49724975 0.49724975 0.49724975 0.49724975 0.49724975 1.40874876
## 23 0.49724975 0.49724975 0.49724975 0.49724975 0.49724975 0.49724975 1.40874876
## 24 0.49724975 0.49724975 0.49724975 0.49724975 0.49724975 0.49724975 1.40874876
## 25 0.29578401 0.29578401 0.18736413 0.29578401 0.29578401 0.29578401 1.40874876
## 26 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 0.49516799
## 27 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 0.56164834
## 28 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 1.40874876 0.56164834
## 15 16 17 18 19 20 21
## 2
## 3
## 4
## 5
## 6
## 7
## 8
## 9
## 10
## 11
## 12
## 13
## 14
## 15
## 16 0.24387639
## 17 1.40874876 1.40874876
## 18 1.40874876 1.40874876 0.12548408
## 19 1.40874876 1.40874876 0.30920822 0.30920822
## 20 1.40874876 1.40874876 0.51155960 0.51155960 0.51155960
## 21 0.56164834 0.56164834 1.40874876 1.40874876 1.40874876 1.40874876
## 22 1.40874876 1.40874876 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876
## 23 1.40874876 1.40874876 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876
## 24 1.40874876 1.40874876 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876
## 25 1.40874876 1.40874876 0.97985304 0.97985304 0.97985304 0.97985304 1.40874876
## 26 0.49516799 0.49516799 1.40874876 1.40874876 1.40874876 1.40874876 0.56164834
## 27 0.56164834 0.56164834 1.40874876 1.40874876 1.40874876 1.40874876 0.46298482
## 28 0.56164834 0.56164834 1.40874876 1.40874876 1.40874876 1.40874876 0.46298482
## 22 23 24 25 26 27
## 2
## 3
## 4
## 5
## 6
## 7
## 8
## 9
## 10
## 11
## 12
## 13
## 14
## 15
## 16
## 17
## 18
## 19
## 20
## 21
## 22
## 23 0.06318240
## 24 0.25154043 0.25154043
## 25 0.49724975 0.49724975 0.49724975
## 26 1.40874876 1.40874876 1.40874876 1.40874876
## 27 1.40874876 1.40874876 1.40874876 1.40874876 0.56164834
## 28 1.40874876 1.40874876 1.40874876 1.40874876 0.56164834 0.23758016
plot(lipids_distBC,cophenetic(lipids_cluster))
cor(lipids_distBC,cophenetic(lipids_cluster),method="spearman")
## [1] 0.8600969
cutree(lipids_cluster,k=4)
## [1] 1 1 1 1 2 2 2 3 3 3 3 3 3 2 2 2 1 1 1 1 4 3 3 3 3 2 4 4
cutree(lipids_cluster,h=0.05)
## [1] 1 2 3 4 5 6 7 8 9 10 11 12 9 13 14 15 16 17 18 19 20 21 22 23 24
## [26] 25 26 27
plot(lipids_cluster, hang = -1, labels = combifac, ylab = "BC")
rect.hclust(lipids_cluster,k=4) # or specify height h instead of k
(aka metric scaling)
Technique to project sites onto Cartesian (Euclidean) coordinates from pairwise dissimilarities.
Imagine a road distance matrix between major towns. A 2D-projection of all pairwise distances produces a map (which gets better with less mountains and straighter roads in the area).
PCoA can be regarded as the inverse operation to computing a distance matrix. However, it can´t produce more “axes” than (n-1) as this is the maximum number of dimensions needed to completely represent n objects. A dissimilarity matrix and a consecutive PCoA are efficient means to condense data with p>>n.
PCoA is an important step in hypothesis-testing methods in the distance domain (see later).
PCoA is based on an eigenvector decomposition like PCA:
# using the same distance matrix as cluster analysis
pcoa<-cmdscale(lipids_distBC,k=2,eig=TRUE,add=TRUE)
# argument add=TRUE means a constant is added to distances to avoid negative eigenvalues
cumsum(pcoa$eig/sum(pcoa$eig)) # contributions of various PCoA axes
## [1] 0.5108797 0.6713554 0.7243661 0.7626502 0.7943804 0.8232193 0.8499307
## [8] 0.8685434 0.8850589 0.8969014 0.9077931 0.9172892 0.9256833 0.9338634
## [15] 0.9419715 0.9496211 0.9562653 0.9622121 0.9677456 0.9731763 0.9782572
## [22] 0.9829655 0.9875741 0.9920859 0.9960663 1.0000000 1.0000000 1.0000000
# first two axes cover 67% of variation of distances
pcoa$points # the site scores (coordinates in reduced space)
## [,1] [,2]
## [1,] 0.039034416 -0.063039145
## [2,] 0.052535090 -0.034222468
## [3,] 0.027399434 -0.087663904
## [4,] -0.009878077 -0.004018133
## [5,] -0.212039581 0.030589392
## [6,] -0.216084698 0.020983290
## [7,] -0.259894649 0.051819890
## [8,] 0.165391223 0.116835950
## [9,] 0.150166208 0.118147706
## [10,] 0.185980907 0.073277130
## [11,] 0.128185960 0.114502815
## [12,] 0.163993524 0.117098404
## [13,] 0.149963125 0.117191012
## [14,] -0.219431512 0.064195917
## [15,] -0.210959910 0.057130086
## [16,] -0.251459219 0.070724865
## [17,] 0.040185127 -0.182894184
## [18,] 0.034791299 -0.174443966
## [19,] 0.036014921 -0.211286639
## [20,] 0.012082138 -0.103177494
## [21,] -0.160218882 -0.026898255
## [22,] 0.202366734 -0.008237486
## [23,] 0.198633241 -0.005774575
## [24,] 0.198199603 -0.068765798
## [25,] 0.194183841 0.040168498
## [26,] -0.233079249 0.023071464
## [27,] -0.078445944 -0.018582519
## [28,] -0.127615070 -0.026731854
col.isolate<-isolate
levels(col.isolate)<-c("white","red")
col.isolate<-as.character(col.isolate)
pch.temperature<-as.numeric(as.character(temperature))
pch.temperature[pch.temperature==6]<-21
pch.temperature[pch.temperature==28]<-23
plot(pcoa$points,pch=pch.temperature,bg=col.isolate)
legend("topleft",pch=c(21,21,23,23),pt.bg=c("white","red","white","red"),
legend=c("6°C - warm isolate","6°C - cold isolate","28°C - warm isolate","28°C - cold isolate"),cex=0.6)
# how to relate species (=fatty acids) to ordination?
wascores(pcoa$points,lip_data) # as weighted averages of site (=sample) scores
text(wascores(pcoa$points,lip_data),labels=names(lip_data),cex=0.5)
ordisurf(pcoa$points,lip_data$FA7_MU,col="darkgreen",add=TRUE) # as contourplot
plot(envfit(pcoa$points,lip_data)) # take care: behaviour of species not necessarily monotonous in ordination spaceAn iterative search for an ordination configuration in a low-dimensional space with inter-object distances representing the observed dissimilarity/distance matrix as well as possible.
The measure of fit used is not actual dissimarity/distance but rank order of dissimilarities.
Iterative procedure:
# non-metric multidimensional scaling (NMDS) #
# note that all resulting axes are equally important, plots may be rotated as needed
# using the same distance matrix as cluster analysis
#mds_lipids<-isoMDS(lipids_distBC, k = 2, maxit=1000)
mds_lipids<-metaMDS(comm=lip_data, distance="bray", k = 2,trymax=100)
## Run 0 stress 0.06823601
## Run 1 stress 0.06291168
## ... New best solution
## ... Procrustes: rmse 0.05036073 max resid 0.1893593
## Run 2 stress 0.07362809
## Run 3 stress 0.06291168
## ... Procrustes: rmse 7.14281e-06 max resid 2.47217e-05
## ... Similar to previous best
## Run 4 stress 0.0682362
## Run 5 stress 0.08036014
## Run 6 stress 0.07362809
## Run 7 stress 0.06291168
## ... Procrustes: rmse 4.44794e-07 max resid 1.629583e-06
## ... Similar to previous best
## Run 8 stress 0.07228103
## Run 9 stress 0.06823697
## Run 10 stress 0.06796483
## Run 11 stress 0.06823625
## Run 12 stress 0.07362809
## Run 13 stress 0.06291168
## ... Procrustes: rmse 1.326353e-05 max resid 3.172257e-05
## ... Similar to previous best
## Run 14 stress 0.07228097
## Run 15 stress 0.06291168
## ... New best solution
## ... Procrustes: rmse 3.589165e-06 max resid 7.27529e-06
## ... Similar to previous best
## Run 16 stress 0.06823585
## Run 17 stress 0.06796483
## Run 18 stress 0.06917428
## Run 19 stress 0.06823585
## Run 20 stress 0.06291168
## ... Procrustes: rmse 4.989635e-06 max resid 1.47675e-05
## ... Similar to previous best
## *** Solution reached
mds_lipids$stress # approximately 6.8% of dissimilarities remain unrepresented
## [1] 0.06291168
mds_lipids$points # the site scores
## MDS1 MDS2
## 1 -0.07281623 -0.042093146
## 2 -0.08834359 -0.016851191
## 3 -0.04897736 -0.075177960
## 4 0.01313624 0.070365419
## 5 0.31339301 0.068577462
## 6 0.31363381 0.088897350
## 7 0.38016017 0.110505209
## 8 -0.24831670 0.131109175
## 9 -0.23159160 0.129025762
## 10 -0.27432580 0.085572655
## 11 -0.20184629 0.128763888
## 12 -0.24619739 0.129699656
## 13 -0.23159160 0.129025757
## 14 0.34897958 -0.018654271
## 15 0.33013239 -0.013738077
## 16 0.40718523 0.009552504
## 17 -0.09652212 -0.204858333
## 18 -0.09263457 -0.201461350
## 19 -0.10685476 -0.291406627
## 20 -0.03121302 -0.120771275
## 21 0.28955893 -0.193173036
## 22 -0.27468239 -0.003772281
## 23 -0.27256597 -0.003232314
## 24 -0.27317859 -0.065288401
## 25 -0.26414048 0.056903940
## 26 0.34983139 0.211431835
## 27 0.11512929 -0.043948014
## 28 0.19465842 -0.055004337
## attr(,"centre")
## [1] TRUE
## attr(,"pc")
## [1] TRUE
## attr(,"halfchange")
## [1] TRUE
## attr(,"internalscaling")
## [1] 3.663216
stressplot(mds_lipids,lipids_distBC)
(gof<-goodness(mds_lipids,statistic="distance")) # goodness of fit by NMDS for each sample
## [1] 0.011062354 0.013697609 0.007607018 0.017971020 0.012576325 0.014203934
## [7] 0.008087433 0.009154932 0.009137032 0.008889532 0.011453452 0.010106705
## [13] 0.009137032 0.010449041 0.011982104 0.013634174 0.011813176 0.011758968
## [19] 0.017537587 0.016442166 0.017081825 0.005768584 0.005300513 0.008717898
## [25] 0.007322299 0.013383859 0.011959534 0.013389617
# plotting of MDS scores
plot(mds_lipids$points,asp=1,xlab="MDS dimension 1",ylab="MDS dimension 2")
points(mds_lipids$points,pch=pch.temperature,bg=col.isolate,cex=gof*200) # to check goodness of fit, large symbol means bad fit
legend("topleft",pch=c(21,21,23,23),pt.bg=c("white","red","white","red"),
legend=c("6°C - warm isolate","6°C - cold isolate","28°C - warm isolate","28°C - cold isolate"),cex=0.7)
# relating underlying variables to ordination in nMDS
wascores(mds_lipids$points,lip_data) # as weighted averages of site (=sample) scores
text(wascores(pcoa$points,lip_data),labels=names(lip_data),cex=0.5)
ordisurf(mds_lipids,lip_data$FA7_MU,col="darkgreen",add=TRUE)
plot(envfit(pcoa$points,lip_data)) # take care: behaviour of species not necessarily monotonous in ordination space
# some more useful graphical tools
plot(mds_lipids$points,pch=pch.temperature,bg=col.isolate,asp=1,xlab="MDS dimension 1",ylab="MDS dimension 2",cex=2)
ordispider(mds_lipids$points,groups=combifac)
ordihull(mds_lipids$points,groups=combifac)
ordiellipse(mds_lipids$points,groups=combifac)
ordicluster(mds_lipids$points,cluster=lipids_cluster)plot(mds_lipids$points,pch=pch.temperature,bg=col.isolate,asp=1,xlab="MDS dimension 1",ylab="MDS dimension 2",cex=2)
ordispider(mds_lipids$points,groups=combifac)
All known study designs with factors or continuous predictors may be transferred to the distance domain.
PERMANOVA: An ANOVA-type of analysis testing effects of factors (also in interaction) on a multivariate matrix, thus MANOVA. Useful to test effects on composition. In the distance domain known as permutational MANOVA. In R implemented as vegan::adonis.
dbRDA (distance-based RDA) and CAP (canonical analysis of principal coordinates): multiple-step analysis to test for effects of continuous predictors on a dissimilarity matrix. In R vegan::capscale.
Mantel-test: An old-school test for correlation between two distance/dissimilarity matrices (e.g. one describing environment or physical distance, the other describing community turnover or genetic differentiation). In R vegan::mantel.
Uses a test statistic based on distances within groups (to a group centroid or averaged among all pairs) versus distances from group centroids to the overall centroid.
Prerequisite similar to ANOVA: homogeneous dispersion (multivariate variance, cloud shape). Tested using within-group distances to centroids.
# PERMANOVA - non-parametric permutational MANOVA #
# a multivariate hypothesis test:
# two factors and 1 multivariate response = "membrane FA composition"
# first testing for homogeneity of dispersion (homogeneous distances to group centroids)
disp.check<-betadisper(lipids_distBC,combifac)
disp.check$distances
## [1] 0.13346669 0.11147756 0.15422004 0.15102228 0.06074253 0.06835717
## [7] 0.05569824 0.03298576 0.01986574 0.05190567 0.02237701 0.03134896
## [13] 0.01877437 0.05215781 0.04500705 0.06406574 0.10040564 0.10078771
## [19] 0.13388533 0.10118002 0.14105434 0.06286176 0.06364213 0.03720424
## [25] 0.09455924 0.15755988 0.06427489 0.02935132
boxplot(disp.check$distances~combifac)
anova(lm(disp.check$distances~combifac))
## Analysis of Variance Table
##
## Response: disp.check$distances
## Df Sum Sq Mean Sq F value Pr(>F)
## combifac 3 0.004967 0.0016556 0.8283 0.4913
## Residuals 24 0.047972 0.0019988
anova(disp.check)
## Analysis of Variance Table
##
## Response: Distances
## Df Sum Sq Mean Sq F value Pr(>F)
## Groups 3 0.004967 0.0016556 0.8283 0.4913
## Residuals 24 0.047972 0.0019988
permutest(disp.check)
##
## Permutation test for homogeneity of multivariate dispersions
## Permutation: free
## Number of permutations: 999
##
## Response: Distances
## Df Sum Sq Mean Sq F N.Perm Pr(>F)
## Groups 3 0.004967 0.0016556 0.8283 999 0.505
## Residuals 24 0.047972 0.0019988
# the actual PERMANOVA
adonis(lipids_distBC~isolate*temperature)
##
## Call:
## adonis(formula = lipids_distBC ~ isolate * temperature)
##
## Permutation: free
## Number of permutations: 999
##
## Terms added sequentially (first to last)
##
## Df SumsOfSqs MeanSqs F.Model R2 Pr(>F)
## isolate 1 0.06599 0.06599 7.988 0.09987 0.004 **
## temperature 1 0.38394 0.38394 46.476 0.58105 0.001 ***
## isolate:temperature 1 0.01258 0.01258 1.522 0.01903 0.193
## Residuals 24 0.19826 0.00826 0.30005
## Total 27 0.66077 1.00000
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1Following scheme taken from description of db-RDA. Essentially 3 steps:
